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Inverse of number triangle A112333.
+20
2
1, -2, 1, 0, -5, 1, 0, 0, -8, 1, 0, 0, 0, -11, 1, 0, 0, 0, 0, -14, 1, 0, 0, 0, 0, 0, -17, 1, 0, 0, 0, 0, 0, 0, -20, 1, 0, 0, 0, 0, 0, 0, 0, -23, 1, 0, 0, 0, 0, 0, 0, 0, 0, -26, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, -29, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -32, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -35, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -38, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
EXAMPLE
Triangle begins
1;
-2,1;
0,-5,1;
0,0,-8,1;
0,0,0,-11,1;
0,0,0,0,-14,1;
0,0,0,0,0,-17,1;
Triangle U, read by rows, where U(n,k) = Product_{j=k..n-1} (3*j+1) for n > k with U(n,n) = 1.
+10
7
1, 1, 1, 4, 4, 1, 28, 28, 7, 1, 280, 280, 70, 10, 1, 3640, 3640, 910, 130, 13, 1, 58240, 58240, 14560, 2080, 208, 16, 1, 1106560, 1106560, 276640, 39520, 3952, 304, 19, 1, 24344320, 24344320, 6086080, 869440, 86944, 6688, 418, 22, 1
COMMENTS
Let G(m, k, p) = (-p)^k*Product_{j=0..k-1}(j - m - 1/p) and T(n, k, p) = G(n-1, n-k, p) then T(n, k, 1) = A094587(n, k), T(n, k, 2) = A112292(n, k) and T(n, k, 3) is this sequence. - Peter Luschny, Jun 01 2009, revised Jun 18 2019
FORMULA
Matrix powers: column 0 of U^(k+1) = column k of A136216 for k >= 0; simultaneously, column k = column 0 of A136216^(3k+1) for k >= 0. Element in column 0, row n, of matrix power U^(k+1) = A007559(n)*C(n+k,k), where A007559 are triple factorials found in column 0 of this triangle.
EXAMPLE
Triangle begins:
1;
1, 1;
4, 4, 1;
28, 28, 7, 1;
280, 280, 70, 10, 1;
3640, 3640, 910, 130, 13, 1;
58240, 58240, 14560, 2080, 208, 16, 1;
1106560, 1106560, 276640, 39520, 3952, 304, 19, 1; ...
Matrix inverse begins:
1;
-1, 1;
0, -4, 1;
0, 0, -7, 1;
0, 0, 0, -10, 1;
0, 0, 0, 0, -13, 1; ...
MAPLE
nmax:=8; for n from 0 to nmax do U(n, n):=1 od: for n from 0 to nmax do for k from 0 to n do if n > k then U(n, k) := mul((3*j+1), j = k..n-1) fi: od: od: for n from 0 to nmax do seq(U(n, k), k=0..n) od: seq(seq(U(n, k), k=0..n), n=0..nmax); # Johannes W. Meijer, Jul 04 2011, revised Nov 23 2012
MATHEMATICA
Table[Product[3*j+1, {j, k, n-1}], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 14 2019 *)
PROG
(PARI) T(n, k)=if(n==k, 1, prod(j=k, n-1, 3*j+1))
(Magma) [[n eq 0 select 1 else k eq n select 1 else (&*[3*j+1: j in [k..n-1]]): k in [0..n]]: n in [0..12]]; // G. C. Greubel, Jun 14 2019 (Sage)
def T(n, k):
if (k==n): return 1
else: return product(3*j+1 for j in (k..n-1))
[[T(n, k) for k in (0..n)] for n in (0..12)] # G. C. Greubel, Jun 14 2019
Triangle T, read by rows, where T(n,k) = A007559(n-k)*C(n,k) where A007559 equals the triple factorials in column 0.
+10
4
1, 1, 1, 4, 2, 1, 28, 12, 3, 1, 280, 112, 24, 4, 1, 3640, 1400, 280, 40, 5, 1, 58240, 21840, 4200, 560, 60, 6, 1, 1106560, 407680, 76440, 9800, 980, 84, 7, 1, 24344320, 8852480, 1630720, 203840, 19600, 1568, 112, 8, 1, 608608000, 219098880, 39836160
COMMENTS
Comments from Peter Bala, Jul 10 2008: (Start) This array is the particular case P(1,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown below n\k|0....................1...............2.........3.....4
----------------------------------------------------------
0..|1.....................................................
1..|a....................1................................
2..|a(a+b)...............2a..............1................
3..|a(a+b)(a+2b).........3a(a+b).........3a........1......
4..|a(a+b)(a+2b)(a+3b)...4a(a+b)(a+2b)...6a(a+b)...4a....1
...
See A094587 for some general properties of these arrays. Other cases recorded in the database include: P(1,0) = Pascal's triangle A007318, P(1,1) = A094587, P(2,0) = A038207, P(3,0) = A027465, P(2,1) = A132159 and P(2,3) = A136216. (End) The generalized Pascal matrix that Bala refers to is itself a special case of application of the formalism of A133314 to fundamental matrices derived from infinitesimal generators described in A133314, of which the fundamental Pascal ( A007318), unsigned Lah ( A105278) and associated Laguerre ( A135278) matrices are special examples. The formalism gives, among other relations, the inverse of T as TI(n,k) = b(n-k)*C(n,k) where the sequence b is given by the list partition transform ( A133314) of A007559; i.e., b = LPT( A007559) = (1,- A008544)= (1,-1,-2,-10,-80,...). The formalism of A132382 may also be applied with the double factorial A001147 replaced by the triple factorial A007559 (see also A133480). - Tom Copeland, Aug 18 2008 Exponential Riordan array [1/(1 - 3*y)^(1/3), y]. The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x).
For example, row 3 is (28, 12, 3, 1) so P(3,x + 1) = (x + 1)*(x + 4)*(x + 7) = 28 + 12*x + 3*x*(x + 3) + x*(x + 3)*(x + 6). (End)
FORMULA
Column k of T = column 0 of U^(k+1) (matrix power) for k>=0 where U = A136214. Matrix square equals A136216, where A136216(n,k) = A008544(n-k)*C(n,k) where A008544 are also triple factorials. T(n,k) = (3*n-3*k-2)*T(n-1,k) + T(n-1,k-1).
E.g.f. exp(x*y)/(1-3*y)^(1/3) = 1 + (1+x)*y + (4+2*x+x^2)*y^2/2! + ... . (End)
EXAMPLE
Column k of T = column 0 of U^(k+1), while
column k of U = column 0 of T^(3k+1) where U = A136214 and column k of V = column 0 of T^(3k+2) where V = A112333. This triangle T begins:
1;
1, 1;
4, 2, 1;
28, 12, 3, 1;
280, 112, 24, 4, 1;
3640, 1400, 280, 40, 5, 1;
58240, 21840, 4200, 560, 60, 6, 1;
1106560, 407680, 76440, 9800, 980, 84, 7, 1; ...
1;
1, 1;
4, 4, 1;
28, 28, 7, 1;
280, 280, 70, 10, 1;
3640, 3640, 910, 130, 13, 1; ...
with triple factorials A007559 in column 0. 1;
2, 1;
10, 5, 1;
80, 40, 8, 1;
880, 440, 88, 11, 1;
12320, 6160, 1232, 154, 14, 1; ...
with triple factorials A008544 in column 0.
MATHEMATICA
T[n_, k_]:= Binomial[n, k]*If[n - k == 0, 1, Product[3*j + 1, {j, 0, n - k - 1}]]; Table[T[n, k], {n, 0, 10}, {k, 0, n}]//Flatten (* G. C. Greubel, Jun 10 2018 *)
PROG
(PARI) T(n, k)=binomial(n, k)*if(n-k==0, 1, prod(j=0, n-k-1, 3*j+1))
Triangle T, read by rows, where T(n,k) = A008544(n-k)*C(n,k) where A008544 equals the triple factorials in column 0.
+10
4
1, 2, 1, 10, 4, 1, 80, 30, 6, 1, 880, 320, 60, 8, 1, 12320, 4400, 800, 100, 10, 1, 209440, 73920, 13200, 1600, 150, 12, 1, 4188800, 1466080, 258720, 30800, 2800, 210, 14, 1, 96342400, 33510400, 5864320, 689920, 61600, 4480, 280, 16, 1
COMMENTS
This array is the particular case P(2,3) of the generalized Pascal triangle P(a,b), a lower unit triangular matrix, shown in the comments to A094587. - Peter Bala, Jul 10 2008 The row polynomials form an Appell sequence. - Tom Copeland, Dec 03 2013
FORMULA
Column k of T = column 0 of V^(k+1) for k>=0 where V = A112333. Equals the matrix square of triangle A136215. T(n,k) = (3*n-3*k-1)*T(n-1,k) + T(n-1,k-1). - Peter Bala, Jul 10 2008 Using the formalism of A132382 modified for the triple rather than the double factorial (replace 2 by 3 in basic formulas), the e.g.f. for the row polynomials is exp(x*t)*(1-3x)^(-2/3). - Tom Copeland, Aug 18 2008 Exponential Riordan array [1/(1 - 3*y)^(2/3), y].
The row polynomials R(n,x) thus form a Sheffer sequence of polynomials with associated delta operator equal to d/dx. Thus d/dx(R(n,x)) = n*R(n-1,x). The Sheffer identity is R(n,x + y) = sum {k = 0..n} binomial(n,k)*y^(n-k)*R(k,x).
Define a polynomial sequence P(n,x) of binomial type by setting P(n,x) = product {k = 0..n-1} (2*x + 3*k) with the convention that P(0,x) = 1. Then this is triangle of connection constants when expressing the basis polynomials P(n,x + 1) in terms of the basis P(n,x). For example, row 3 is (80, 30, 6, 1) so P(3,x + 1) = (2*x + 2)*(2*x + 5)*(2*x + 8) = 80 + 20*(2*x) + 6*(2*x*(2*x + 3)) + (2*x)*(2*x + 3)*(2*x + 6). (End)
EXAMPLE
Triangle begins:
1;
2, 1;
10, 4, 1;
80, 30, 6, 1;
880, 320, 60, 8, 1;
12320, 4400, 800, 100, 10, 1;
209440, 73920, 13200, 1600, 150, 12, 1;
4188800, 1466080, 258720, 30800, 2800, 210, 14, 1; ...
MATHEMATICA
(* The function RiordanArray is defined in A256893. *)
PROG
(PARI) {T(n, k) = binomial(n, k)*if(n-k==0, 1, prod(j=0, n-k-1, 3*j+2))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
a(n) = Product_{k in M_n} k, M_n = {k | 1 <= k <= 3n and k mod 3 = n mod 3}.
+10
1
1, 1, 10, 162, 280, 12320, 524880, 1106560, 96342400, 7142567040, 17041024000, 2324549427200, 254561089305600, 664565853952000, 126757680265216000, 18763697892715776000, 52580450364682240000, 13106744139423334400000, 2480410751833883860992000
COMMENTS
For n > 0:
a(3*n) = A032031(3*n) = 3^(3*n) * Gamma(3*n + 1). a(3*n-1) = A008544(3*n-1) = 3^(3*n-1) * Gamma(3*n - 1/3) / Gamma(2/3). a(3*n+1) = A007559(3*n+1) = 3^(3*n+3/2) * Gamma(3*n + 4/3) * Gamma(2/3) / (2*Pi).
FORMULA
a(3*n+3)/(a(3*n)*a(3)) = A006566(n+1); Dodecahedral numbers a(3*n+4)/a(3*n+1) = A136214(3*n+4, 3*n+1) a(3*n+5)/a(3*n+2) = A112333(3*n+5, 3*n+2) (End)
MAPLE
A190903 := proc(n) local k; mul(k, k = select(k-> k mod 3 = n mod 3, [$1 .. 3*n])) end: seq( A190903(n), n=0..17);
MATHEMATICA
a[n_] := Switch[Mod[n, 3], 0, 3^n Gamma[n+1], 2, 3^n Gamma[n+2/3]/ Gamma[2/3], 1, 3^(n-1) Gamma[n+1/3]/Gamma[4/3]] // Round;
PROG
(PARI) a(n) = prod(k=1, 3*n, if (k % 3 == n % 3, k, 1)); \\ Michel Marcus, Jun 25 2019 and May 14 2020
1, 3, 16, 129, 1420, 19881, 337978, 6759561, 155469904, 4042217505, 117224307646, 3751177844673, 131291224563556, 4989066533415129, 204551727870020290, 9000276026280892761, 423012973235201959768, 21150648661760097988401, 1120984379073285193385254
FORMULA
a(n) = (3*n-1) * a(n-1) + 1.
PROG
(PARI) a008544(n) = prod(k=1, n, 3*k-1);
a(n) = a008544(n)*sum(k=0, n, 1/a008544(k));
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