A001469 -id:A001469 - cp's OEIS frontend

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

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

A083007 a(n) = Sum_{k=0..n-1} 3^kB(k)C(n,k) where B(k) is the k-th Bernoulli number and C(n,k)=binomial(n,k).

Original entry on oeis.org

0, 1, -2, 1, 4, -5, -26, 49, 328, -809, -6710, 20317, 201772, -722813, -8370194, 34607305, 457941136, -2145998417, -31945440878, 167317266613, 2767413231220, -16020403322021, -291473080313162, 1848020950359841, 36679231132772824, -252778977216700025, -5435210060467425446

Offset: 0

Benoit Cloitre, May 31 2003

Seiichi Manyama, Table of n, a(n) for n = 0..510
G. Almkvist and A. Meurman, Values of Bernoulli polynomials and Hurwitz's Zeta function at rational points, C. R. Math. Rep. Acad. Sci. Canada 13 (1991), 104-109.
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.
B. Sury, The value of Bernoulli Polynomials at rational numbers, Bull. London Math. Soc. 25 (1993), 327-29.

Cf. A001469.
Cf. A036968, A083008, A083009, A083010, A083011, A083012, A083013, A083014.
Cf. A002111.

A083007 := proc(n)
    3*x/(1+exp(x)+exp(2*x)) ;
    coeftayl(%,x=0,n) ;
    %*n! ;
end proc:
seq(A083007(n),n=0..30) ; # R. J. Mathar, Jul 13 2023

Range[0, 15]! CoefficientList[ Series[ 3x/(1 + Exp[x] + Exp[ 2x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Table[Sum[3^k BernoulliB[k]Binomial[n,k],{k,0,n-1}],{n,0,30}] (* Harvey P. Dale, May 26 2014 *)

a(n)=sum(k=0,n-1,3^k*binomial(n,k)*bernfrac(k))

E.g.f.: 3x/(1+e^x+e^(2x)). - Ira M. Gessel, Jan 28 2012
From Peter Bala, Mar 01 2015: (Start)
a(2*n+1) = (-1)^(n+1)*A002111(n) for n >= 1.
a(n) = 3^n * ( B(n,1/3) - B(n,0) ), where B(n,x) denotes the n-th Bernoulli polynomial. More generally, Almkvist and Meurman show that k^n * ( B(n, 1/k) - B(n, 0) ) is an integer sequence for k = 2,3,4,..., which proves the integrality of A083008 through A083014.
a(0) = 1 and for n >= 1, a(n) = 1 - 1/(n + 1)*Sum_{k = 1..n-1} 3^(n-k)*binomial(n+1,k)*a(k) (Sury, Section 1). (End)

A083008 a(n) = Sum_{k=0..n-1} 4^kB(k)C(n,k) where B(k) is the k-th Bernoulli number and C(n,k) = binomial(n,k).

Original entry on oeis.org

0, 1, -3, 3, 9, -25, -99, 427, 2193, -12465, -79515, 555731, 4247577, -35135945, -313193811, 2990414715, 30461046561, -329655706465, -3777604994187, 45692713833379, 581778811909545, -7777794952988025, -108933009112011843, 1595024111042171723, 24370176181315498929

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..485

Cf. A001469.
Bisection is A009843.
Cf. A036968, A083007, A083009, A083010, A083011, A083012, A083013, A083014.

Range[0, 15]! CoefficientList[ Series[ 4x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Table[Sum[4^k*BernoulliB[k] Binomial[n, k], {k, 0, n - 1}], {n, 0, 24}] (* Michael De Vlieger, Sep 28 2016 *)

a(n)=sum(k=0,n-1,4^k*binomial(n,k)*bernfrac(k))

E.g.f.: 4*x/(1+exp(x)+exp(2*x)+exp(3*x)). - Ira M. Gessel, Feb 23 2012

a(n) ~ n! * (cos(n*Pi/2)-sin(n*Pi/2)) * 2^(n+1) / Pi^n. - Vaclav Kotesovec, Mar 02 2014

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)(-1)^(m-i)i^(n-m)*i!, for n >= 2, m = 1..n-1.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 13, 13, 1, 1, 29, 73, 29, 1, 1, 61, 301, 301, 61, 1, 1, 125, 1081, 2069, 1081, 125, 1, 1, 253, 3613, 11581, 11581, 3613, 253, 1, 1, 509, 11593, 57749, 95401, 57749, 11593, 509, 1, 1, 1021, 36301, 268381, 673261, 673261, 268381, 36301, 1021, 1

Offset: 2

N. J. A. Sloane, May 07 2016

Gives number of bitriangular permutations. Could be prefixed with row 0 containing a single 1. - N. J. A. Sloane, Jan 10 2018

			Triangle begins:
n\m  [1]     [2]     [3]     [4]     [5]     [6]     [7]     [8]
[2]  1;
[3]  1,      1;
[4]  1,      5,      1;
[5]  1,     13,     13,      1;
[6]  1,     29,     73,     29,      1;
[7]  1,     61,    301,    301,     61,      1;
[8]  1,    125,   1081,   2069,   1081,    125,      1;
[9]  1,    253,   3613,  11581,  11581,   3613,    253,      1;
...

Gheorghe Coserea, Rows n = 2..101, flattened
F. Alayont and N. Krzywonos, Rook Polynomials in Three and Higher Dimensions, 2012.
Beáta Bényi, A Bijection for the Boolean Numbers of Ferrers Graphs, Graphs and Combinatorics (2022) Vol. 38, No. 10.
Beata Bényi and Peter Hajnal, Combinatorial properties of poly-Bernoulli relatives, arXiv preprint arXiv:1602.08684 [math.CO], 2016. See D_{n,k}.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, Duke Mathematical Journal 13.2 (1946): 259-268. The array is on page 267.
Irving Kaplansky and John Riordan, The problem of the rooks and its applications, in Combinatorics, Duke Mathematical Journal, 13.2 (1946): 259-268. [Annotated scanned copy]
D. E. Knuth, Parades and poly-Bernoulli bijections, Mar 31 2024. See (16.2).
D. E. Knuth, Notes on four arrays of numbers arising from the enumeration of CRC constraints and min-and-max-closed constraints, May 06 2024. Mentions this sequence.
J. Riordan, Letter to N. J. A. Sloane, Dec. 1976.

Column 2 is A036563.
Largest term in each row gives A272645.
Second diagonal from the right is 2^i - 3.
Third diagonal from the right edge is A006230.
T(2n,n) gives A048144.
For row sums see A297195.
Cf. A008277, A001469, A371761.

A272644 := proc(n,m)
    add(combinat[stirling2](m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!,i=0..m) ;
end proc:
seq(seq(A272644(n,m),m=1..n-1),n=2..10) ; # R. J. Mathar, Mar 04 2018

Table[Sum[StirlingS2[m + 1, i + 1] (-1)^(m - i) i^(n - m) i!, {i, 0, m} ], {n, 11}, {m, n - 1}] /. {} -> {0} // Flatten  (* Michael De Vlieger, May 19 2016 *)

A(n,m) = sum(i=0, m, stirling(m+1, i+1, 2) * (-1)^((m-i)%2) * i^(n - m) * i!);
concat(vector(10, n, vector(n, m, A(n+1, m))))  \\ Gheorghe Coserea, May 16 2016

T(n,m) = Sum_{i=0..m} Stirling2(m+1, i+1)*(-1)^(m-i)*i^(n-m)*i!, for n>=2, m=1..n-1, where Stirling2(n,k) is defined by A008277.

A001469(n+1) = Sum_{m=1..2*n-1} (-1)^(m-1)*T(2*n,m). - Gheorghe Coserea, May 18 2016

More terms from Gheorghe Coserea, May 16 2016

A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

Original entry on oeis.org

0, 1, 1, 33, 437, 22205, 978873, 81005113, 7356832669, 949918117653, 142805534055905, 27120922891214801, 6016195462632487941, 1592800634594574194413, 486576430503128985793417, 171866951067212728072402665, 69025662074064538734826793453

Offset: 0

Ilya Gutkovskiy, Dec 21 2017

nonn

			tan(x*tan(x/2)) = x^2/2! + x^4/4! + 33*x^6/6! + 437*x^8/8! + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..200

Cf. A000182, A001469, A003718, A009707, A024265, A110501, A296835, A296841, A296842.

nmax = 16; Table[(CoefficientList[Series[Tan[x Tan[x/2]], {x, 0, 2 nmax}], x] Range[0, 2 nmax]!)[[n]], {n, 1, 2 nmax + 1, 2}]

a(n) = (2*n)! * [x^(2*n)] tan(x*tan(x/2)).

a(n) ~ c * d^n * n^(2*n + 1/2) / exp(2*n), where d = 16/Pi^2 = 1.621138938277404343102071411355642222469740394755... is the root of the equation tan(1/sqrt(d)) = Pi*sqrt(d)/4 and c = 1.75568815831... - Vaclav Kotesovec, Dec 21 2017, updated Mar 16 2024

A083009 a(n) = Sum_{k=0,n-1} 5^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

Original entry on oeis.org

0, 1, -4, 6, 16, -74, -264, 1946, 9056, -88434, -512024, 6154786, 42716496, -607884394, -4920817384, 80834386026, 747784582336, -13923204233954, -144898927180344, 3015393801263666, 34867899296006576, -801997872697905114, -10201104981227536904, 256982712667627683706

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..467

Cf. A001469.

Cf. A036968, A083007, A083008, A083010, A083011, A083012, A083013, A083014.

Range[0, 15]! CoefficientList[ Series[ 5x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Table[Sum[5^k*BernoulliB[k] Binomial[n, k], {k, 0, n - 1}], {n, 0, 23}] (* Michael De Vlieger, Sep 28 2016 *)

a(n)=sum(k=0,n-1,5^k*binomial(n,k)*bernfrac(k))

E.g.f.: 5x/(1+exp(x)+exp(2x)+exp(3x)+exp(4x)). - Benoit Cloitre, Oct 26 2012 (following I. Gessel).

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

A083010 a(n) = 6^n(B_n(1/6)-B_n(0)) where B_n(x) is the n-th Bernoulli polynomial.

Original entry on oeis.org

0, 1, -5, 10, 25, -170, -575, 6370, 28225, -415826, -2294975, 41649850, 275622625, -5922729722, -45718037855, 1134081384850, 10004182986625, -281284596509858, -2791456543622015, 87722769712529770, 967282878165054625, -33597252908389628234, -407509096583935700255

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..452

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083011, A083012, A083013, A083014.

Range[0, 15]! CoefficientList[ Series[ 6x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)

a(n)=sum(k=0,n-1,6^k*binomial(n,k)*bernfrac(k))

{a(n)=if(n<1, 0, n!*polcoeff( 6*x*(exp(x+x*O(x^n))-1)/(exp(6*x +x*O(x^n))-1), n))} /* Michael Somos, Aug 02 2006 */

E.g.f.: 6x(exp(x)-1)/(exp(6x)-1). - Michael Somos, Aug 02 2006

a(n) = Sum_{k=0..n-1} 6^k*B(k)*C(n,k) where B(k) is the k-th Bernoulli number and C(n,k) = binomial(n,k).

cp's OEIS Frontend

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

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A083007 a(n) = Sum_{k=0..n-1} 3^k*B(k)*C(n,k) where B(k) is the k-th Bernoulli number and C(n,k)=binomial(n,k).

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A083008 a(n) = Sum_{k=0..n-1} 4^k*B(k)*C(n,k) where B(k) is the k-th Bernoulli number and C(n,k) = binomial(n,k).

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A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)*(-1)^(m-i)*i^(n-m)*i!, for n >= 2, m = 1..n-1.

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A296839 Expansion of e.g.f. tan(x*tan(x/2)) (even powers only).

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A083009 a(n) = Sum_{k=0,n-1} 5^k*B(k)*binomial(n,k) where B(k) is the k-th Bernoulli number.

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A083010 a(n) = 6^n(B_n(1/6)-B_n(0)) where B_n(x) is the n-th Bernoulli polynomial.

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A083007 a(n) = Sum_{k=0..n-1} 3^kB(k)C(n,k) where B(k) is the k-th Bernoulli number and C(n,k)=binomial(n,k).

A083008 a(n) = Sum_{k=0..n-1} 4^kB(k)C(n,k) where B(k) is the k-th Bernoulli number and C(n,k) = binomial(n,k).

A272644 Triangle read by rows: T(n,m) = Sum_{i=0..m} Stirling2(m+1,i+1)(-1)^(m-i)i^(n-m)*i!, for n >= 2, m = 1..n-1.

A083009 a(n) = Sum_{k=0,n-1} 5^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

A083011 a(n) = Sum_{k=0..n-1} 7^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

A083012 a(n) = Sum_{k=0..n-1} 8^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

A083013 a(n) = Sum_{k=0..n-1} 9^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.