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

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

Offset: 1

Author

Don Knuth, N. J. A. Sloane

Keywords

• nonn
• easy
• nice

Comments

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

Examples

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

References

• 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.

Links

• 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.

Crossrefs

Cf. A001469, A005439, A130168, A130169.

First column, first diagonal and row sums of triangle A014784.

Also row sums of triangle A239894.

Programs

Mathematica

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. *)

PARI

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

PARI

{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)

PARI

{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

Python

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

Sage

# 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

Formula

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

Extensions

More terms from David W. Wilson, Jan 11 2001

Edited by Ralf Stephan, Apr 17 2004