|
|
A001469
|
|
Genocchi numbers (of first kind); unsigned coefficients give expansion of x*tan(x/2).
(Formerly M3041 N1233)
|
|
76
|
|
|
-1, 1, -3, 17, -155, 2073, -38227, 929569, -28820619, 1109652905, -51943281731, 2905151042481, -191329672483963, 14655626154768697, -1291885088448017715, 129848163681107301953, -14761446733784164001387, 1884515541728818675112649, -268463531464165471482681379
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,3
|
|
COMMENTS
|
The Genocchi numbers satisfy Seidel's recurrence: for n>1, 0 = Sum_{j=0..[n/2]} C(n,2j)*a(n-j). - Ralf Stephan, Apr 17 2004
The (n+1)st Genocchi number is the number of Dumont permutations of the first kind on 2n letters. In a Dumont permutation of the first kind, each even integer must be followed by a smaller integer and each odd integer is either followed by a larger integer or is the last element. - Ralf Stephan, Apr 26 2004
According to Hetyei [2017], "alternation acyclic tournaments in which at least one ascent begins at each vertex, except for the largest one, are counted by the Genocchi numbers of the first kind." - Danny Rorabaugh, Apr 25 2017
|
|
REFERENCES
|
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
L. Euler, Institutionum Calculi Differentialis, volume 2 (1755), para. 181.
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. 73.
A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani, Ann. Sci. Mat. Fis., 3 (1852), 395-405.
R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 528.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = 2*(1-4^n)*B_{2n} (B = Bernoulli numbers).
x*tan(x/2) = Sum_{n>=1} x^(2*n)*abs(a(n))/(2*n)! = x^2/2 + x^4/24 + x^6/240 + 17*x^8/40320 + 31*x^10/725760 + O(x^11).
E.g.f.: 2*x/(1 + exp(x)) = x + Sum_{n>=1} a(2*n)*x^(2*n)/(2*n)! = -x^2/2! + x^4/4! - 3 x^6/6! + 17 x^8/8! + ...
O.g.f.: Sum_{n>=0} n!^2*(-x)^(n+1) / Product_{k=1..n} (1-k^2*x). - Paul D. Hanna, Jul 21 2011
a(n) = Sum_{k=0..2n-1} 2^k*B(k)*binomial(2*n,k) where B(k) is the k-th Bernoulli number. - Benoit Cloitre, May 31 2003
G.f.: -x/(1+x/(1+2x/(1+4x/(1+6x/(1+9x/(1+12x/(1+16x/(1+20x/(1+25x/(1+...(continued fraction). - Philippe Deléham, Nov 22 2011
E.g.f.: E(x) = 2*x/(exp(x)+1) = x*(1-(x^3+2*x^2)/(2*G(0)-x^3-2*x^2)); G(k) = 8*k^3 + (12+4*x)*k^2 + (4+6*x+2*x^2)*k + x^3 + 2*x^2 + 2*x - 2*(x^2)*(k+1)*(2*k+1)*(x+2*k)*(x+2*k+4)/G(k+1); (continued fraction, Euler's kind, 1-step). - Sergei N. Gladkovskii, Jan 18 2012
a(n) = (-1)^n*(2*n)!*Pi^(-2*n)*4*(1-4^(-n))*Li{2*n}(1). - Peter Luschny, Jun 29 2012
Asymptotic: abs(a(n)) ~ 8*Pi*(2^(2*n)-1)*(n/(Pi*exp(1)))^(2*n+1/2)*exp(1/2+(1/24)/n-(1/2880)/n^3+(1/40320)/n^5+...). - Peter Luschny, Jul 24 2013
G.f.: x/(T(0)-x) -1, where T(k) = 2*x*k^2 + 4*x*k + 2*x - 1 - x*(-1+x+2*x*k+x*k^2)*(k+2)^2/T(k+1); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2013
G.f.: -1 + x/(T(0)+x), where T(k) = 1 + (k+1)*(k+2)*x/(1+x*(k+2)^2/T(k+1)); (continued fraction). - Sergei N. Gladkovskii, Nov 17 2013
|
|
MAPLE
|
A001469 := proc(n::integer) (2*n)!*coeftayl( 2*x/(exp(x)+1), x=0, 2*n) end proc:
|
|
MATHEMATICA
|
Table[4 n PolyLog[1 - 2 n, -1], {n, 1, 19}] (* Peter Luschny, Aug 17 2021 *)
|
|
PROG
|
(PARI) a(n)=if(n<1, 0, n*=2; 2*(1-2^n)*bernfrac(n))
(PARI) {a(n)=polcoeff(sum(m=0, n, m!^2*(-x)^(m+1)/prod(k=1, m, 1-k^2*x+x*O(x^n))), n)} /* Paul D. Hanna, Jul 21 2011 */
(Sage) # Algorithm of L. Seidel (1877)
# n -> [a(1), ..., a(n)] for n >= 1.
D = [0]*(n+2); D[1] = -1
R = []; b = False
for i in(0..2*n-1) :
h = i//2 + 1
if b :
for k in range(h-1, 0, -1) : D[k] -= D[k+1]
else :
for k in range(1, h+1, 1) : D[k] -= D[k-1]
b = not b
if not b : R.append(D[h])
return R
(Magma) [2*(1 - 4^n) * Bernoulli(2*n): n in [1..25]]; // Vincenzo Librandi, Oct 15 2018
(Python)
from sympy import bernoulli
|
|
CROSSREFS
|
|
|
KEYWORD
|
sign,easy,nice
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|