%I M2049 N0811 #312 Apr 19 2024 18:58:08

%S 1,1,2,12,288,34560,24883200,125411328000,5056584744960000,

%T 1834933472251084800000,6658606584104736522240000000,

%U 265790267296391946810949632000000000,127313963299399416749559771247411200000000000,792786697595796795607377086400871488552960000000000000

%N Superfactorials: product of first n factorials.

%C a(n) is also the Vandermonde determinant of the numbers 1,2,...,(n+1), i.e., the determinant of the (n+1) X (n+1) matrix A with A[i,j] = i^j, 1 <= i <= n+1, 0 <= j <= n. - Ahmed Fares (ahmedfares(AT)my-deja.com), May 06 2001

%C a(n) = (1/n!) * D(n) where D(n) is the determinant of order n in which the (i,j)-th element is i^j. - _Amarnath Murthy_, Jan 02 2002

%C Determinant of S_n where S_n is the n X n matrix S_n(i,j) = Sum_{d|i} d^j. - _Benoit Cloitre_, May 19 2002

%C Appears to be det(M_n) where M_n is the n X n matrix with m(i,j) = J_j(i) where J_k(n) denote the Jordan function of row k, column n (cf. A059380(m)). - _Benoit Cloitre_, May 19 2002

%C a(2n+1) = 1, 12, 34560, 125411328000, ... is the Hankel transform of A000182 (tangent numbers) = 1, 2, 16, 272, 7936, ...; example: det([1, 2, 16, 272; 2, 16, 272, 7936; 16, 272, 7936, 353792; 272, 7936, 353792, 22368256]) = 125411328000. - _Philippe Deléham_, Mar 07 2004

%C Determinant of the (n+1) X (n+1) matrix whose i-th row consists of terms 1 to n+1 of the Lucas sequence U(i,Q), for any Q. When Q=0, the Vandermonde matrix is obtained. - _T. D. Noe_, Aug 21 2004

%C Determinant of the (n+1) X (n+1) matrix A whose elements are A(i,j) = B(i+j) for 0 <= i,j <= n, where B(k) is the k-th Bell number, A000110(k) [I. Mezo, JIS 14 (2011) # 11.1.1]. - _T. D. Noe_, Dec 04 2004

%C The Hankel transform of the sequence A090365 is A000178(n+1); example: det([1,1,3; 1,3,11; 3,11,47]) = 12. - _Philippe Deléham_, Mar 02 2005

%C Theorem 1.3, page 2, of Polynomial points, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6, provides an example of an Abelian quotient group of order (n-1) superfactorial, for each positive integer n. The quotient is obtained from sequences of polynomial values. - E. F. Cornelius, Jr. (efcornelius(AT)comcast.net), Apr 09 2007

%C Starting with offset 1 this is a 'Matryoshka doll' sequence with alpha=1, the multiplicative counterpart to the additive A000292. seq(mul(mul(i,i=alpha..k), k=alpha..n),n=alpha..12). - _Peter Luschny_, Jul 14 2009

%C For n>0, a(n) is also the determinant of S_n where S_n is the n X n matrix, indexed from 1, S_n(i,j)=sigma_i(j), where sigma_k(n) is the generalized divisor sigma function: A000203 is sigma_1(n). - _Enrique Pérez Herrero_, Jun 21 2010

%C a(n) is the multiplicative Wiener index of the (n+1)-vertex path. Example: a(4)=288 because in the path on 5 vertices there are 3 distances equal to 2, 2 distances equal to 3, and 1 distance equal to 4 (2*2*2*3*3*4=288). See p. 115 of the Gutman et al. reference. - _Emeric Deutsch_, Sep 21 2011

%C a(n-1) = Product_{j=1..n-1} j! = V(n) = Product_{1 <= i < j <= n} (j - i) (a Vandermondian V(n), see the Ahmed Fares May 06 2001 comment above), n >= 1, is in fact the determinant of any n X n matrix M(n) with entries M(n;i,j) = p(j-1,x = i), 1 <= i, j <= n, where p(m,x), m >= 0, are monic polynomials of exact degree m with p(0,x) = 1. This is a special x[i] = i choice in a general theorem given in Vein-Dale, p. 59 (written for the transposed matrix M(n;j,x_i) = p(i-1,x_j) = P_i(x_j) in Vein-Dale, and there a_{k,k} = 1, for k=1..n). See the Aug 26 2013 comment under A049310, where p(n,x) = S(n,x) (Chebyshev S). - _Wolfdieter Lang_, Aug 27 2013

%C a(n) is the number of monotonic magmas on n elements labeled 1..n with a symmetric multiplication table. I.e., Product(i,j) >= max(i,j); Product(i,j) = Product(j,i). - _Chad Brewbaker_, Nov 03 2013

%C The product of the pairwise differences of n+1 integers is a multiple of a(n) [and this does not hold for any k > a(n)]. - _Charles R Greathouse IV_, Aug 15 2014

%C a(n) is the determinant of the (n+1) X (n+1) matrix M with M(i,j) = (n+j-1)!/(n+j-i)!, 1 <= i <= n+1, 1 <= j <= n+1. - _Stoyan Apostolov_, Aug 26 2014

%C All terms are in A064807 and all terms after a(2) are in A005101. - _Ivan N. Ianakiev_, Sep 02 2016

%C Empirical: a(n-1) is the determinant of order n in which the (i,j)-th entry is the (j-1)-th derivative of x^(x+i-1) evaluated at x=1. - _John M. Campbell_, Dec 13 2016

%C Empirical: If f(x) is a smooth, real-valued function on an open neighborhood of 0 such that f(0)=1, then a(n) is the determinant of order n+1 in which the (i,j)-th entry is the (j-1)-th derivative of f(x)/((1-x)^(i-1)) evaluated at x=0. - _John M. Campbell_, Dec 27 2016

%C Also the automorphism group order of the n-triangular honeycomb rook graph. - _Eric W. Weisstein_, Jul 14 2017

%C Is the zigzag Hankel transform of A000182. That is, a(2*n+1) is the Hankel transform of A000182 and a(2*n+2) is the Hankel transform of A000182(n+1). - _Michael Somos_, Mar 11 2020

%C Except for n = 0, 1, superfactorial a(n) is never a square (proof in link Mabry and Cormick, FFF 4 p. 349); however, when k belongs to A349079 (see for further information), there exists m, 1 <= m <= k such that a(k) / m! is a square. - _Bernard Schott_, Nov 29 2021

%D Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 545.

%D Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 135-145.

%D A. Fletcher, J. C. P. Miller, L. Rosenhead and L. J. Comrie, An Index of Mathematical Tables. Vols. 1 and 2, 2nd ed., Blackwell, Oxford and Addison-Wesley, Reading, MA, 1962, Vol. 1, p. 50.

%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 231.

%D H. J. Ryser, Combinatorial Mathematics. Mathematical Association of America, Carus Mathematical Monograph 14, 1963, p. 53.

%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D R. Vein and P. Dale, Determinants and Their Applications in Mathematical Physics, Springer, 1999.

%H Boris Hostnik, <a href="/A000178/b000178.txt">Table of n, a(n) for n = 0..46</a>

%H Christian Aebi and Grant Cairns, <a href="http://www.jstor.org/stable/10.4169/amer.math.monthly.122.5.433">Generalizations of Wilson's Theorem for Double-, Hyper-, Sub-and Superfactorials</a>, The American Mathematical Monthly 122.5 (2015): 433-443.

%H Andreas B. G. Blobel, <a href="https://arxiv.org/abs/2203.09519">On convolution powers of 1/x</a>, arXiv:2203.09519 [math.CO], 2022.

%H E. F. Cornelius, Jr. and Phill Schultz, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL10/Schultz/schultz14.html">Polynomial points </a>, Journal of Integer Sequences, Vol. 10 (2007), Article 07.3.6.

%H Selden Crary, <a href="http://arxiv.org/abs/1406.6326">Factorization of the Determinant of the Gaussian-Covariance Matrix of Evenly Spaced Points Using an Inter-dimensional Multiset Duality</a>, arXiv preprint arXiv:1406.6326 [math.ST], 2014-2019.

%H N. Destainville, R. Mosseri and F. Bailly, <a href="http://dx.doi.org/10.1007/BF02181243">Configurational Entropy of Codimension-One Tilings and Directed Membranes</a>, J. Stat. Phys. 87, Nos 3/4, 697 (1997).

%H J. East and R. D. Gray, <a href="http://arxiv.org/abs/1404.2359">Idempotent generators in finite partition monoids and related semigroups</a>, arXiv preprint arXiv:1404.2359 [math.GR], 2014.

%H Richard Ehrenborg, <a href="http://www.jstor.org/stable/2589352">The Hankel determinant of exponential polynomials</a>, Amer. Math. Monthly, 107 (2000), 557-560.

%H William Q. Erickson and Jan Kretschmann, <a href="https://arxiv.org/abs/2311.07522">The structure and normalized volume of Monge polytopes</a>, arXiv:2311.07522 [math.CO], 2023. See p. 7.

%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/glshkn/glshkn.html">Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for A002109, A000178) [Broken link]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010622230958/http://www.mathsoft.com/asolve/constant/glshkn/glshkn.html">Glaisher-Kinkelin Constant</a> (gives asymptotic expressions for A002109, A000178) [From the Wayback machine]

%H Ivan Gutman, Wolfgang Linert, István Lukovits and Željko Tomović, <a href="http://dx.doi.org/10.1021/ci990060s">The multiplicative version of the Wiener index</a>, J. Chem. Inf. Comput. Sci., Vol. 40, No. 1 (2000), pp. 113-116.

%H Brady Haran and Sophie Maclean, <a href="https://www.youtube.com/watch?v=xV4A8oU3yew">What's special about 288?</a>, Numberphile video (2023).

%H Aoife Hennessy, <a href="http://repository.wit.ie/1693/1/AoifeThesis.pdf">A Study of Riordan Arrays with Applications to Continued Fractions, Orthogonal Polynomials and Lattice Paths</a>, Ph. D. Thesis, Waterford Institute of Technology, Oct. 2011.

%H Nick Hobson, <a href="/A000178/a000178.py.txt">Python program for this sequence</a>.

%H A. M. Ibrahim, <a href="http://www.nntdm.net/papers/nntdm-19/NNTDM-19-2-30_42.pdf">Extension of factorial concept to negative numbers</a>, Notes on Number Theory and Discrete Mathematics, Vol. 19, 2013, 2, 30-42.

%H Pavel L. Krapivsky, Jean-Marc Luck and Kirone Mallick, <a href="http://doi.org/10.1088/1742-5468/aaa79a">Quantum return probability of a system of N non-interacting lattice fermions</a>, Journal of Statistical Mechanics: Theory and Experiment, Vol. 2018, No. 2 (2018), 023104; <a href="https://arxiv.org/abs/1710.08178">arXiv preprint</a>, arXiv:1710.08178 [cond-mat.mes-hall], 2017-2018.

%H Jeffrey C. Lagarias and Harsh Mehta, <a href="https://doi.org/10.1142/S1793042116500044">Products of binomial coefficients and unreduced Farey fractions</a>, International Journal of Number Theory, Vol. 12, No. 1 (2016), pp. 57-91; <a href="http://arxiv.org/abs/1409.4145">arXiv preprint</a>, arXiv:1409.4145 [math.NT], 2014-2015.

%H Mogens Esrom Larsen, <a href="http://www.maa.org/programs/faculty-and-departments/classroom-capsules-and-notes/wronskian-harmony">Wronskian Harmony</a>, Mathematics Magazine, vol. 63, no. 1, 1990, pp. 33-37.

%H John W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.

%H Rick Mabry and Laura McCormick, <a href="https://www.austms.org.au/wp-content/uploads/Gazette/2009/Nov09/TechPaperMabry.pdf">Square products of punctured sequences of factorials</a>, Gaz. Aust. Math. Soc., 2009.

%H Rémy Mosseri and Francis Bailly, <a href="http://dx.doi.org/10.1142/S0217979293002419">Configurational Entropy in Octagonal Tiling Models</a>, Int. J. Mod. Phys. B, Vol. 7, No. 6-7 (1993), pp. 1427-1436.

%H Rémy Mosseri, F. Bailly and C. Sire, <a href="http://dx.doi.org/10.1016/0022-3093(93)90342-U">Configurational Entropy in Random Tiling Models</a>, J. Non-Cryst. Solids, Vol. 153-154 (1993), pp. 201-204.

%H Amarnath Murthy, <a href="http://vixra.org/abs/1403.0675">Miscellaneous Results and Theorems on Smarandache terms and factor partitions</a>, Smarandache Notions Journal, Vol. 11, No. 1-2-3, Spring 2000.

%H Amarnath Murthy and Charles Ashbacher, <a href="http://www.gallup.unm.edu/~smarandache/MurthyBook.pdf">Generalized Partitions and Some New Ideas on Number Theory and Smarandache Sequences</a>, Hexis, Phoenix; USA 2005. See Section 3.14.

%H Christian Radoux, <a href="https://www.ams.org/journals/notices/197804/197804FullIssue.pdf">Query 145</a>, Notices Amer. Math. Soc., 25-3 (1978), p. 197.

%H Christian Radoux, <a href="http://www.mat.univie.ac.at/~slc/opapers/s28radoux.html">Déterminants de Hankel et théorème de Sylvester</a>, Séminaire Lotharingien de Combinatoire, B28b (1992), 9 pp.

%H Vignesh Raman, <a href="https://arxiv.org/abs/2012.00882">The Generalized Superfactorial, Hyperfactorial and Primorial functions</a>, arXiv:2012.00882 [math.NT], 2020.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BellNumber.html">Bell Number</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/FactorialProducts.html">Factorial Products</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/GraphAutomorphism.html">Graph Automorphism</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/LucasSequence.html">Lucas Sequence</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Superfactorial.html">Superfactorial</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/VandermondeDeterminant.html">Vandermonde Determinant</a>.

%H <a href="/index/Di#divseq">Index to divisibility sequences</a>.

%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%H <a href="/index/O#Olympiads">Index to sequences related to Olympiads and other Mathematical competitions</a>.

%F a(0) = 1, a(n) = n!*a(n-1). - _Lee Hae-hwang_, May 13 2003, corrected by _Ilya Gutkovskiy_, Jul 30 2016

%F a(0) = 1, a(n) = 1^n * 2^(n-1) * 3^(n-2) * ... * n = Product_{r=1..n} r^(n-r+1). - _Amarnath Murthy_, Dec 12 2003 [Formula corrected by _Derek Orr_, Jul 27 2014]

%F a(n) = sqrt(A055209(n)). - _Philippe Deléham_, Mar 07 2004

%F a(n) = Product_{i=1..n} Product_{j=0..i-1} (i-j). - _Paul Barry_, Aug 02 2008

%F log a(n) = 0.5*n^2*log n - 0.75*n^2 + O(n*log n). - _Charles R Greathouse IV_, Jan 13 2012

%F Asymptotic: a(n) ~ exp(zeta'(-1) - 3/4 - (3/4)*n^2 - (3/2)*n)*(2*Pi)^(1/2 + (1/2)*n)*(n+1)^((1/2)*n^2 + n + 5/12). For example, a(100) is approx. 0.270317...*10^6941. (See A213080.) - _Peter Luschny_, Jun 23 2012

%F G.f.: 1 + x/(U(0) - x) where U(k) = 1 + x*(k+1)! - x*(k+2)!/U(k+1); (continued fraction). - _Sergei N. Gladkovskii_, Oct 02 2012

%F G.f.: G(0)/2, where G(k) = 1 + 1/(1 - 1/(1 + 1/((k+1)!*x*G(k+1)))); (continued fraction). - _Sergei N. Gladkovskii_, Jun 14 2013

%F G.f.: 1 = Sum_{n>=0} a(n)*x^n / Product_{k=1..n+1} (1 + k!*x). - _Paul D. Hanna_, Oct 02 2013

%F A203227(n+1)/a(n) -> e, as n -> oo. - _Daniel Suteu_, Jul 30 2016

%F From _Ilya Gutkovskiy_, Jul 30 2016: (Start)

%F a(n) = G(n+2), where G(n) is the Barnes G-function.

%F a(n) ~ exp(1/12 - n*(3*n+4)/4)*n^(n*(n+2)/2 + 5/12)*(2*Pi)^((n+1)/2)/A, where A is the Glaisher-Kinkelin constant (A074962).

%F Sum_{n>=0} (-1)^n/a(n) = A137986. (End)

%F 0 = a(n)*a(n+2)^3 + a(n+1)^2*a(n+2)^2 - a(n+1)^3*a(n+3) for all n in Z (if a(-1)=1). - _Michael Somos_, Mar 11 2020

%F Sum_{n>=0} 1/a(n) = A287013 = 1/A137987. - _Amiram Eldar_, Nov 19 2020

%F a(n) = Wronskian(1, x, x^2, ..., x^n). - _Mohammed Yaseen_, Aug 01 2023

%F From _Andrea Pinos_, Apr 04 2024: (Start)

%F a(n) = e^(Sum_{k=1..n} (Integral_{x=1..k+1} Psi(x) dx)).

%F a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + x*Psi(x)) dx).

%F a(n) = e^(Integral_{x=1..n+1} (log(sqrt(2*Pi)) - (x-1/2) + (n+1)*Psi(x) - log(Gamma(x))) dx).

%F Psi(x) is the digamma function. (End)

%e a(3) = (1/6)* | 1 1 1 | 2 4 8 | 3 9 27 |

%e a(7) = n! * a(n-1) = 7! * 24883200 = 125411328000.

%e a(12) = 1! * 2! * 3! * 4! * 5! * 6! * 7! * 8! * 9! * 10! * 11! * 12!

%e = 1^12 * 2^11 * 3^10 * 4^9 * 5^8 * 6^7 * 7^6 * 8^5 * 9^4 * 10^3 * 11^2 * 12^1

%e = 2^56 * 3^26 * 5^11 * 7^6 * 11^2.

%e G.f. = 1 + x + 2*x^2 + 12*x^3 + 288*x^4 + 34560*x^5 + 24883200*x^6 + ...

%p A000178 := proc(n)

%p mul(i!,i=1..n) ;

%p end proc:

%p seq(A000178(n),n=0..10) ; # _R. J. Mathar_, Oct 30 2015

%t a[0] := 1; a[1] := 1; a[n_] := n!*a[n - 1]; Table[a[n], {n, 1, 12}] (* _Stefan Steinerberger_, Mar 10 2006 *)

%t Table[BarnesG[n], {n, 2, 14}] (* _Zerinvary Lajos_, Jul 16 2009 *)

%t FoldList[Times,1,Range[20]!] (* _Harvey P. Dale_, Mar 25 2011 *)

%t RecurrenceTable[{a[n] == n! a[n - 1], a[0] == 1}, a, {n, 0, 12}] (* _Ray Chandler_, Jul 30 2015 *)

%t BarnesG[Range[2, 20]] (* _Eric W. Weisstein_, Jul 14 2017 *)

%o (PARI) A000178(n)=prod(k=2,n,k!) \\ _M. F. Hasler_, Sep 02 2007

%o (PARI) a(n)=polcoeff(1-sum(k=0, n-1, a(k)*x^k/prod(j=1, k+1, (1+j!*x+x*O(x^n)) )), n) \\ _Paul D. Hanna_, Oct 02 2013

%o (PARI) for(j=1,13, print1(prod(k=1,j,k^(j-k)),", ")) \\ _Hugo Pfoertner_, Apr 09 2020

%o (Maxima) A000178(n):=prod(k!,k,0,n)$ makelist(A000178(n),n,0,30); /* _Martin Ettl_, Oct 23 2012 */

%o (Ruby)

%o def mono_choices(a,b,n)

%o n - [a,b].max

%o end

%o def comm_mono_choices(n)

%o accum =1

%o 0.upto(n-1) do |i|

%o i.upto(n-1) do |j|

%o accum = accum * mono_choices(i,j,n)

%o end

%o end

%o accum

%o end

%o 1.upto(12) do |k|

%o puts comm_mono_choices(k)

%o end # _Chad Brewbaker_, Nov 03 2013

%o (Magma) [&*[Factorial(k): k in [0..n]]: n in [0..20]]; // _Bruno Berselli_, Mar 11 2015

%o (Python)

%o A000178_list, n, m = [1], 1,1

%o for i in range(1,100):

%o m *= i

%o n *= m

%o A000178_list.append(n) # _Chai Wah Wu_, Aug 21 2015

%o (Python)

%o from math import prod

%o def A000178(n): return prod(i**(n-i+1) for i in range(2,n+1)) # _Chai Wah Wu_, Nov 26 2023

%Y Partial products of A000142.

%Y Cf. A002109, A036561, A000292, A098694, A098695, A113271, A087316, A113208, A113231, A113257, A113258, A113320, A113336, A113498, A113173, A113170, A113475, A113492, A113497, A113533, A113534, A113535, A113153, A113154, A113122, A114045, A055462, A137986, A137987.

%Y A002109(n)*A000178(n-1) = (n!)^n = A036740(n) for n >= 1.

%Y A000178 is the Hankel transform (see A001906 for definition) of A000085, A000110, A000296, A005425, A005493, A005494 and A045379. - _John W. Layman_, Jul 28 2000

%Y Cf. A255322, A255358, A255359, A255360.

%Y Cf. A051675, A255321, A255323, A255344, A287013.

%Y Cf. A348692, A349079.

%K nonn,nice,easy

%O 0,3

%A _N. J. A. Sloane_