A211183 -id:A211183 - cp's OEIS frontend

A110501 Unsigned Genocchi numbers (of first kind) of even index.

1, 1, 3, 17, 155, 2073, 38227, 929569, 28820619, 1109652905, 51943281731, 2905151042481, 191329672483963, 14655626154768697, 1291885088448017715, 129848163681107301953, 14761446733784164001387, 1884515541728818675112649, 268463531464165471482681379

Offset: 1

Michael Somos, Jul 23 2005

The Genocchi numbers satisfy Seidel's recurrence: for n > 1, 0 = Sum_{j=0..floor(n/2)} (-1)^j*binomial(n, 2*j)*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
The (n+1)-st Genocchi number is also the number of ways to place n rooks (attacking along planes; also called super rooks of power 2 by Golomb and Posner) on the three-dimensional Genocchi boards of size n. The Genocchi board of size n consists of cells of the form (i, j, k) where min{i, j} <= k and 1 <= k <= n. A rook placement on this board can also be realized as a pair of permutations of n the smallest number in the i-th position of the two permutations is not larger than i. - Feryal Alayont, Nov 03 2012
The (n+1)-st Genocchi number is also the number of Dumont permutations of the second kind, third kind, and fourth kind on 2n letters. In a Dumont permutation of the second kind, all odd positions are weak excedances and all even positions are deficiencies. In a Dumont permutation of the third kind, all descents are from an even value to an even value. In a Dumont permutation of the fourth kind, all deficiencies are even values at even positions. - Alexander Burstein, Jun 21 2019
The (n+1)-st Genocchi number is also the number of semistandard Young tableaux of skew shape (n+1,n,...,1)/(n-1,n-2,...,1) such that the entries in row i are at most i for i=1,...,n+1. - Alejandro H. Morales, Jul 26 2020
The (n+1)-st Genocchi number is also the number of positive terms of the Okounkov-Olshanski formula for the number of standard tableaux of skew shape (n+1,n,n-1,...,1)/(n-1,n-2,...,1), given by the (2n+1)-st Euler number A000111. - Alejandro H. Morales, Jul 26 2020
The (n+1)-st Genocchi number is also the number of collapsed permutations in (2n-1) letters. A permutation pi of size 2n-1 is said to be collapsed if ceil(k/2) <= pi^{-1}(k) <= n + floor(k/2). There are 3 collapsed permutations of size 3, namely 123, 132 and 213. - Arvind Ayyer, Oct 23 2020

			E.g.f.: x*tan(x/2) = x^2/2! + x^4/4! + 3*x^6/6! + 17*x^8/8! + 155*x^10/10! + ...
O.g.f.: A(x) = x + x^2 + 3*x^3 + 17*x^4 + 155*x^5 + 2073*x^6 + ...
where A(x) = x + x^2/(1+x) + 2!^2*x^3/((1+x)*(1+4*x)) + 3!^2*x^4/((1+x)*(1+4*x)*(1+9*x)) + 4!^2*x^5/((1+x)*(1+4*x)*(1+9*x)*(1+16*x)) + ... . - _Paul D. Hanna_, Jul 21 2011
From _Gary W. Adamson_, Jul 19 2011: (Start)
The first few rows of production matrix M are:
  1, 2,  0,  0,  0, 0, ...
  1, 3,  3,  0,  0, 0, ...
  1, 4,  6,  4,  0, 0, ...
  1, 5, 10, 10,  5, 0, ...
  1, 6, 15, 20, 15, 6, ... (End)

L. Carlitz, A conjecture concerning Genocchi numbers. Norske Vid. Selsk. Skr. (Trondheim) 1971, no. 9, 4 pp. MR0297697 (45 #6749)
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
Leonhard Euler, Institutionum Calculi Differentialis, volume 2 (1755), para. 181.
A. Genocchi, Intorno all'espressione generale de'numeri Bernulliani, Ann. Sci. Mat. Fis., 3 (1852), 395-405.
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2 (1999) p. 74; see Problem 5.8.

Alan Sokal, Table of n, a(n) for n = 1..250 (terms up to a(100) from Alois P. Heinz)
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, to appear in Involve
F. Alayont, R. Moger-Reischer and R. Swift, Rook Number Interpretations of Generalized Central Factorial and Genocchi Numbers, preprint, 2012.
R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., 1 (1945), pp. 385-386.
A. Ayyer, D. Hathcock and P. Tetali, Toppleable Permutations, Excedances and Acyclic Orientations, arXiv:2010.11236 [math.CO], 2020.
Peter Bala, A triangle for calculating the Genocchi numbers
Ange Bigeni, A bijection between the irreducible k-shapes and the surjective pistols of height k-1, arXiv preprint arXiv:1402.1383 [math.CO] (2014). Also Discrete Math., 338 (2015), 1432-1448.
Ange Bigeni, Enumerating the symplectic Dellac configurations, arXiv:1705.03804 [math.CO], 2017.
Ange Bigeni, The universal sl2 weight system and the Kreweras triangle, arXiv:1712.05475 [math.CO], 2017.
Ange Bigeni, Combinatorial interpretations of the Kreweras triangle in terms of subset tuples, arXiv:1712.01929 [math.CO], 2017.
Ange Bigeni, A generalization of the Kreweras triangle through the universal sl_2 weight system, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 309-326.
A. Burstein, M. Josuat-Vergès and W. Stromquist, New Dumont permutations, Pure Math. Appl. (Pu.M.A.) 21 (2010), no. 2, 177-206.
E. Clark and R. Ehrenborg, The excedance algebra, Discr. Math., 313 (2013), 1429-1435.
Bishal Deb and Alan D. Sokal, Classical continued fractions for some multivariate polynomials generalizing the Genocchi and median Genocchi numbers, arXiv:2212.07232 [math.CO], 2022. See pp. 4, 12.
Bishal Deb, Continued fractions using a Laguerre digraph interpretation of the Foata-Zeilberger bijection and its variants, arXiv:2304.14487 [math.CO], 2023. See pp. 4, 11.
D. Dumont, Sur une conjecture de Gandhi concernant les nombres de Genocchi, Discrete Mathematics 1 (1972) 321-327.
D. Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., 41 (1974), 305-318.
Dominique Dumont and Dominique Foata, Une propriété de symétrie des nombres de Genocchi Bull. Soc. Math. France 104 (1976), no. 4, 433-451. MR0434830 (55 #7794)
D. Dumont and G. Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Preprint, Annotated scanned copy.
Dominique Dumont and Gérard Viennot, A combinatorial interpretation of the Seidel generation of Genocchi numbers, Combinatorial mathematics, optimal designs and their applications (Proc. Sympos. Combin. Math. and Optimal Design, Colorado State Univ., Fort Collins, Colo., 1978). Ann. Discrete Math. 6 (1980), 77-87. MR0593524 (82j:10024).
A. L. Edmonds and S, Klee, The combinatorics of hyperbolized manifolds, arXiv:1210.7396 [math.CO], 2012.
Richard Ehrenborg and Einar Steingrímsson, Yet another triangle for the Genocchi numbers, European J. Combin. 21 (2000), no. 5, 593-600. MR1771988 (2001h:05008)
Sen-Peng Eu, Tung-Shan Fu, Hsin-Hao Lai, and Yuan-Hsun Lo, Gamma-positivity for a Refinement of Median Genocchi Numbers, arXiv:2103.09130 [math.CO], 2021.
Vincent Froese and Malte Renken, Terrain-like Graphs and the Median Genocchi Numbers, arXiv:2210.16281 [math.CO], 2022.
J. M. Gandhi, Research Problems: A Conjectured Representation of Genocchi Numbers, Amer. Math. Monthly 77 (1970), no. 5, 505-506. MR1535914
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.
S. W. Golomb and E. C. Posner, Rook Domains, Latin Squares, Affine Planes, and Error-Distributing Codes, Transactions of the Information Theory Group of the IEEE, Vol. 10, No. 3 (1964), 196-208.
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Guo-Niu Han and Jing-Yi Liu, Combinatorial proofs of some properties of tangent and Genocchi numbers, European Journal of Combinatorics, Vol. 71 (2018), pp. 99-110; arXiv preprint, arXiv:1707.08882 [math.CO], 2017-2018.
Florent Hivert and Olivier Mallet, Combinatorics of k-shapes and Genocchi numbers, in FPSAC 2011, Reykjavik, Iceland DMTCS proc. AO, 2011, 493-504.
Alexander Lazar and Michelle L. Wachs, The Homogenized Linial Arrangement and Genocchi Numbers, arXiv:1910.07651 [math.CO], 2019.
Zhicong Lin and Sherry H.F. Yan, Cycles on a multiset with only even-odd drops, arXiv:2108.03790 [math.CO], 2021. See also Disc. Math. (2022) Vol. 345, No. 2, 112683.
A. H. Morales and D. G. Zhu, On the Okounkov--Olshanski formula for standard tableaux of skew shape, arXiv:2007.05006 [math.CO], 2020.
Qiongqiong Pan and Jiang Zeng, Cycles of even-odd drop permutations and continued fractions of Genocchi numbers, arXiv:2108.03200 [math.CO], 2021.
John Riordan and Paul R. Stein, Proof of a conjecture on Genocchi numbers, Discrete Math. 5 (1973), 381-388. MR0316372 (47 #4919) - From _N. J. A. Sloane_, Jun 12 2012
Alan Sokal, Table of n, a(n) for n = 1..10000 [315 MB file]
H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 66-67.

Cf. A000182, A001469, A002105, A005439, A036968, A211183, A297703.

[Abs(2*(4^n-1)*Bernoulli(2*n)): n in [1..20]]; // Vincenzo Librandi, Jul 28 2017

A110501 := proc(n)
    2*(-1)^n*(1-4^n)*bernoulli(2*n) ;
end proc:
seq(A110501(n),n=0..10) ; # R. J. Mathar, Aug 02 2013

a[n_] := 2*(4^n - 1) * BernoulliB[2n] // Abs; Table[a[n], {n, 19}] (* Jean-François Alcover, May 23 2013 *)

{a(n) = if( n<1, 0, 2 * (-1)^n * (1 - 4^n) * bernfrac( 2*n))};

{a(n) = if( n<1, 0, (2*n)! * polcoeff( x * tan(x/2 + x * O(x^(2*n))), 2*n))};

{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 */

upto(n) = my(v1, v2, v3); v1 = vector(n, i, 0); v1[1] = 1; v2 = vector(n-1, i, ((i+1)^2)\4); v3 = v1; for(i=2, n, for(j=2, i-1, v1[j] += v2[i-j+1]*v1[j-1]); v1[i] = v1[i-1]; v3[i] = v1[i]); v3 \\ Mikhail Kurkov, Aug 28 2025

from sympy import bernoulli
def A110501(n): return ((2<<(m:=n<<1))-2)*abs(bernoulli(m)) # Chai Wah Wu, Apr 14 2023

# Algorithm of L. Seidel (1877)
# n -> [a(1), ..., a(n)] for n >= 1.
def A110501_list(n) :
    D = []; [D.append(0) for i in (0..n+2)]; D[1] = 1
    R = [] ; b = True
    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 b : R.append(D[h])
    return R
A110501_list(19) # Peter Luschny, Apr 01 2012

[2*(-1)^n*(1-4^n)*bernoulli(2*n) for n in (1..20)] # G. C. Greubel, Nov 28 2018

(-1)^n * a(n) = A036968(2*n) = A001469(n).
a(n) = 2*(-1)^n*(1-4^n)*B_{2*n} (B = A027641/A027642 are Bernoulli numbers).
A002105(n) = 2^(n-1)/n * a(n). - Don Knuth, Jan 16 2007
A000111(2*n-1) = a(n)*2^(2*n-2)/n. - Alejandro H. Morales, Jul 26 2020
E.g.f.: x * tan(x/2) = Sum_{k > 0} a(k) * x^(2*k) / (2*k)!.
E.g.f.: x * tan(x/2) = x^2 / (2 - x^2 / (6 - x^2 / (... 4*k+2 - x^2 / (...)))). - Michael Somos, Mar 13 2014
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..2*n} (-1)^(n-k+1)*Stirling2(2*n, k)*A059371(k). - Vladeta Jovovic, Feb 07 2004
O.g.f.: A(x) = x/(1-x/(1-2*x/(1-4*x/(1-6*x/(1-9*x/(1-12*x/(... -[(n+1)/2]*[(n+2)/2]*x/(1- ...)))))))) (continued fraction). - Paul D. Hanna, Jan 16 2006
a(n) = Pi^(-2*n)*integral(log(t/(1-t))^(2*n)-log(1-1/t)^(2*n) dt,t=0,1). - Gerry Martens, May 25 2011
a(n) = the upper left term of M^(n-1); M is an infinite square production matrix with M[i,j] = C(i+1,j-1), i.e., Pascal's triangle without the first two rows and right border, see the examples and Maple program. - Gary W. Adamson, Jul 19 2011
G.f.: 1/U(0) where U(k) = 1 + 2*(k^2)*x - x*((k+1)^2)*(x*(k^2)+1)/U(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Sep 15 2012
a(n+1) = Sum_{k=0..n} A211183(n, k)*2^(n-k). - Philippe Deléham, Feb 03 2013
G.f.: 1 + x/(G(0)-x) where G(k) =  2*x*(k+1)^2 + 1 - x*(k+2)^2*(x*k^2+2*x*k+x+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Feb 10 2013
G.f.: G(0) where G(k) = 1 + x*(2*k+1)^2/( 1 + x + 4*x*k + 4*x*k^2 - 4*x*(k+1)^2*(1 + x + 4*x*k + 4*x*k^2)/(4*x*(k+1)^2 + (1 + 4*x + 8*x*k + 4*x*k^2)/G(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Feb 11 2013
G.f.: R(0), where R(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 1/(1 - x*(k+1)*(k+2)/( x*(k+1)*(k+2) - 1/R(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 27 2013
E.g.f. (offset 1): sqrt(x)*tan(sqrt(x)/2) = Q(0)*x/2, where Q(k) = 1 - x/(x - 4*(2*k+1)*(2*k+3)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Jan 06 2014
Pi^2/6 = 2*Sum_{k=1..N} (-1)^(k-1)/k^2 + (-1)^N/N^2(1 - 1/N + 1/N^3 - 3/N^5 + 17/N^7 - 155/N^9 +- ...), where the terms in the parenthesis are (-1)^n*a(n)/N^(2n-1). - M. F. Hasler, Mar 11 2015
a(n) = 2*n*|euler(2*n-1, 0)|. - Peter Luschny, Jun 09 2016
a(n) = 4^(1-n) * (4^n-1) * Pi^(-2*n) * (2*n)! * zeta(2*n). - Daniel Suteu, Oct 14 2016
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+...). [Given in A001469 by Peter Luschny, Jul 24 2013, copied May 14 2022.]
a(n) = A000182(n) * n / 4^(n-1) (Han and Liu, 2018). - Amiram Eldar, May 17 2024

Edited by M. F. Hasler, Mar 22 2015

A000366 Genocchi numbers of second kind (A005439) divided by 2^(n-1).

Original entry on oeis.org

1, 1, 2, 7, 38, 295, 3098, 42271, 726734, 15366679, 391888514, 11860602415, 420258768950, 17233254330343, 809698074358250, 43212125903877439, 2599512037272630686, 175079893678534943287, 13122303354155987156306, 1088559958829010054171343, 99456043127935948731527942

Offset: 1

Don Knuth, N. J. A. Sloane

The earliest known reference to these numbers is the Dellac Marseille memoir. - Don Knuth, Jul 11 2007
According to Ira Gessel, Dellac's interpretation is the following: start with a 2n X n array of cells and consider the set D of cells in rows i through i+n of column i, for i from 1 to n. Then a(n) is the number of subsets of D containing two cells in each column and one cell in each row.
Barsky proved that for even n>1, a(n) is congruent to 3 mod 4 and for odd n>1, congruent to 2 mod 4. Gessel shows that for even n>5, a(n) is congruent to 4n-1 mod 16 and for odd n>2 that a(n)/2 is congruent to 2-n mod 8.
The entry for A005439 has further information.
The number of sequences (I_1,...,I_{n-1}) consisting of subsets of the set {1,...,n} such that the number of elements in I_k is exactly k and I_k\subset I_{k+1}\cup {k+1}. The Euler characteristics of the degenerate flag varieties of type A. - Evgeny Feigin, Dec 15 2011
Kreweras proved that for n>2, a(n) is alternatively congruent to 2 and to 7 mod 36. - Michel Marcus, Nov 06 2012

			G.f. = x + x^2 + 2*x^3 + 7*x^4 + 38*x^5 + 295*x^6 + 3098*x^7 + ...

Anonymous, L'Intermédiaire des Mathématiciens, 7 (1900), p. 328.
Hippolyte Dellac, Problem 1735, L'Intermédiaire des Mathématiciens, Vol. 7 (1900), p. 9 ff.
E. Lemoine, L'Intermédiaire des Mathématiciens, 8 (1901), 168-169.
L. Seidel, Über eine einfache Entstehungsweise der Bernoulli'schen Zahlen und einiger verwandten Reihen, Sitzungsberichte der mathematisch-physikalischen Classe der königlich bayerischen Akademie der Wissenschaften zu München, volume 7 (1877), 157-187.

T. D. Noe, Table of n, a(n) for n=1..100
D. Barsky, Congruences pour les nombres de Genocchi de 2e espèce, Groupe d'étude d'Analyse ultramétrique, 8e année, no. 34, 1980/81, 13 pp.
Ange Bigeni, Enumerating the symplectic Dellac configurations, arXiv:1705.03804 [math.CO], 2017.
Ange Bigeni, The universal sl2 weight system and the Kreweras triangle, arXiv:1712.05475 [math.CO], 2017.
Ange Bigeni, Combinatorial interpretations of the Kreweras triangle in terms of subset tuples, arXiv:1712.01929 [math.CO], 2017.
Ange Bigeni, A generalization of the Kreweras triangle through the universal sl_2 weight system, Journal of Combinatorial Theory, Series A (2019) Vol. 161, 309-326.
Ange Bigeni and Evgeny Feigin, Symmetric Dellac configurations, arXiv:1808.04275 [math.CO], 2018.
Alin Bostan, Lucia Di Vizio, and Kilian Raschel, Singular walks in the quarter plane and Bernoulli numbers, arXiv:2504.13542 [math.CO], 2025. See p. 29.
Hippolyte Dellac, Note sur l'élimination, méthode de parallélogramme, Annales de la Faculté des Sciences de Marseille, XI (1901), 141-164.
E. Feigin, Degenerate flag varieties and the median Genocchi numbers, arXiv:1101.1898 [math.AG], 2011.
E. Feigin, The median Genocchi numbers, Q-analogues and continued fractions, arXiv:1111.0740 [math.CO], 2011-2012.
Vincent Froese and Malte Renken, Terrain-like Graphs and the Median Genocchi Numbers, arXiv:2210.16281 [math.CO], 2022.
I. M. Gessel, Applications of the classical umbral calculus, arXiv:math/0108121 [math.CO], 2001.
G. Han and J. Zeng, On a q-sequence that generalizes the median Genocchi numbers, Annal Sci. Math. Quebec, 23(1999), no. 1, 63-72
G. Kreweras, Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce, Europ. J. Comb., vol. 18, pp. 49-58, 1997.
G. Kreweras and D. Dumont, Sur les anagrammes alternés, Discrete Mathematics, Volume 211, Issues 1-3, 28 January 2000, Pages 103-110.
Qiongqiong Pan and Jiang Zeng, Cycles of even-odd drop permutations and continued fractions of Genocchi numbers, arXiv:2108.03200 [math.CO], 2021.

Cf. A001469, A005439, A130168, A130169.
First column, first diagonal and row sums of triangle A014784.
Also row sums of triangle A239894.

a[n_] = (-2^(-1))^(n-2)* Sum[ Binomial[n, k]*(1 - 2^(n+k+1))*BernoulliB[n+k+1], {k, 0, n}]; Table[a[n], {n,19}] (* Jean-François Alcover, Jul 18 2011, after PARI prog. *)

a(n)=(-1/2)^(n-2)*sum(k=0,n,binomial(n,k)*(1-2^(n+k+1))*bernfrac(n+k+1))

{a(n)=local(CF=1+x*O(x^n));if(n<1,return(0), for(k=1,n,CF=1/(1-((n-k)\2+1)*((n-k)\2+2)/2*x*CF));return(Vec(CF)[n]))} (Hanna)

{a(n)=polcoeff( x*sum(m=0, n, m!*(m+1)!*(x/2)^m / prod(k=1, m,1 + k*(k+1)*x/2 +x*O(x^n)) ), n)} \\ Paul D. Hanna, Feb 03 2013

from math import comb
from sympy import bernoulli
def A000366(n): return (-1 if n&1 else 1)*sum(comb(n,k)*(1-(1<>n-2 if n>1 else 1 # Chai Wah Wu, Apr 14 2023

# Algorithm of L. Seidel (1877)
# n -> [a(1), ..., a(n)] for n >= 1.
def A000366_list(n) :
    D = [0]*(n+2); D[1] = 1
    R = []; z = 1/2; 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]
            z *= 2
        else :
            for k in range(1, h+1, 1) :  D[k] += D[k-1]
        b = not b
        if not b : R.append(D[1]/z)
    return R
A000366_list(19) # Peter Luschny, Jun 29 2012

From Don Knuth, Jul 11 2007: (Start)
The anonymous 1900 note in Interm. Math. gives a formula that is equivalent to a nice generating function:
For example, the first four terms on the right are
1
... 2x - 2x^2 +  2x^3 + ...
........ 9x^2 - 36x^3 + ...
............... 72x^3 + ...
summing to 1 + 2x + 7x^2 + 38x^3 + ... . Of course one can replace x with 2x and get a generating function for A005439. (End)
(-2)^(2-n) * Sum_{k=0..n} C(n, k)*(1-2^(n+k+1))*B(n+k+1), with B(n) the Bernoulli numbers.
O.g.f.: A(x) = x/(1-x/(1-x/(1-3*x/(1-3*x/(1-6*x/(1-6*x/(... -[n/2+1]*[n/2+2]/2*x/(1- ...)))))))) (continued fraction). - Paul D. Hanna, Oct 07 2005
Sum_{n>0} a(n)x^n = Sum_{n>0} (n!^2/2^{n-1}) (x^n/((1+x)(1+3x)...(1+binomial(n,2)x))).
a(n+1) = Sum_{k=0..n} A211183(n,k). - Philippe Deléham, Feb 03 2013
G.f.: Q(0)*2 - 2, where Q(k) = 1 - x*(k+1)^2/( x*(k+1)^2 - 2/(1 - x*(k+1)^2/( x*(k+1)^2 - 2/Q(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Oct 22 2013
a(n) ~ 2^(n+5) * n^(2*n+3/2) / (exp(2*n) * Pi^(2*n+1/2)). - Vaclav Kotesovec, Oct 28 2014

More terms from David W. Wilson, Jan 11 2001

Edited by Ralf Stephan, Apr 17 2004

cp's OEIS Frontend

A110501 Unsigned Genocchi numbers (of first kind) of even index.

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A000366 Genocchi numbers of second kind (A005439) divided by 2^(n-1).

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A221972 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (2*k-1) / (1 + k*(2*k-1)*x).

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A211194 G.f.: Sum_{n>=0} n! * (x/2)^n * Product_{k=1..n} (3*k-1) / (1 + k*(3*k-1)/2*x).

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PARI

Formula

A221972 G.f.: Sum_{n>=0} n! * x^n * Product_{k=1..n} (2k-1) / (1 + k(2k-1)x).

A211194 G.f.: Sum_{n>=0} n! * (x/2)^n * Product_{k=1..n} (3k-1) / (1 + k(3k-1)/2x).