A083014 -id:A083014 - cp's OEIS frontend

A036968 Genocchi numbers (of first kind): expansion of 2*x/(exp(x)+1).

1, -1, 0, 1, 0, -3, 0, 17, 0, -155, 0, 2073, 0, -38227, 0, 929569, 0, -28820619, 0, 1109652905, 0, -51943281731, 0, 2905151042481, 0, -191329672483963, 0, 14655626154768697, 0, -1291885088448017715, 0, 129848163681107301953

Offset: 1

N. J. A. Sloane

The sign of a(1) depends on which convention one chooses: B(n) = B_n(1) or B(n) = B_n(0) where B(n) are the Bernoulli numbers and B_n(x) the Bernoulli polynomials (see the Wikipedia article on Bernoulli numbers). The definition given is in line with B(n) = B_n(0). The convention B(n) = B_n(1) corresponds to the e.g.f. -2*x/(1+exp(-x)). - Peter Luschny, Jun 28 2013
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
Named after the Italian mathematician Angelo Genocchi (1817-1889). - Amiram Eldar, Jun 06 2021
Conjecture: For any positive integer n, -a(n+1) is the permanent of the n X n matrix M with M(j, k) = floor((2*j - k)/n), (j,k=1..n). - Zhi-Wei Sun, Sep 07 2021
A corresponding conjecture can also be made for L. Seidel's 'Genocchi numbers of second kind' A005439. - Peter Luschny, Sep 08 2021

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
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.
Ronald L. Graham, Donald E. Knuth and Oren Patashnik, Concrete Mathematics. Addison-Wesley, Reading, MA, 1990, p. 528.
Richard P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 5.8.

Seiichi Manyama, Table of n, a(n) for n = 1..551
Shreya Ahirwar, Susanna Fishel, Parikshita Gya, Pamela E. Harris, Nguyen Pham, Andrés R. Vindas-Meléndez, and Dan Khanh Vo, Maximal Chains in Bond Lattices, Elec. J. Combinatorics (2022) Vol. 29, No. 3, #P3.11.
R. C. Archibald, Review of Terrill-Terrill paper, Math. Comp., Vol. 1, No. 10 (1945), pp. 385-386.
Fatima Zohra Bensaci, Rachid Boumahdi, and Laala Khaldi, Finite Sums Involving Fibonacci and Lucas Numbers, J. Int. Seq. (2024). See p. 2.
Beáta Bényi and Matthieu Josuat-Vergès, Combinatorial proof of an identity on Genocchi numbers, arXiv:2010.10060 [math.CO], 2020.
Kwang-Wu Chen, An Interesting Lemma for Regular C-fractions, J. Integer Seqs., Vol. 6 (2003), Article 03.4.8.
Dominique Dumont, Interpretations combinatoires des nombres de Genocchi, Duke Math. J., Vol. 41 (1974), pp. 305-318.
Dominique Dumont, Interprétations combinatoires des nombres de Genocchi, Duke Math. J., Vol. 41 (1974), pp. 305-318. (Annotated scanned copy)
Shishuo Fu, Zhicong Lin and Zhi-Wei Sun, Proofs of five conjectures relating permanents to combinatorial sequences, arXiv:2109.11506 [math.CO], 2021.
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.
Hans-Christian Herbig, Daniel Herden and Christopher Seaton, On compositions with x^2/(1-x), Proceedings of the American Mathematical Society, Vol. 143, No. 11 (2015), pp. 4583-4596; arXiv preprint, arXiv:1404.1022 [math.SG], 2014.
Gábor Hetyei, Alternation acyclic tournaments, European Journal of Combinatorics, Vol. 81 (2019), pp. 1-21; arXiv preprint, arXiv:math/1704.07245 [math.CO], 2017.
G. Kreweras, Sur les permutations comptées par les nombres de Genocchi de 1-ière et 2-ième espèce, Europ. J. Comb., Vol. 18 (1997), pp. 49-58.
Ali Lavasani and Sagar Vijay, The Stability of Gapped Quantum Matter and Error-Correction with Adiabatic Noise, arXiv:2402.14906 [cond-mat.str-el], 2024. See p. 16.
Chellal Redha, An Identity for Generalized Euler Polynomials, arXiv:2402.17063 [math.NT], 2024. See p. 7.
Zhi-Wei Sun, Arithmetic properties of some permanents, arXiv:2108.07723 [math.GM], 2021.
H. M. Terrill and E. M. Terrill, Tables of numbers related to the tangent coefficients, J. Franklin Inst., 239 (1945), 66-67.
Wikipedia, Bernoulli number.
Wikipedia, Genocchi number.

A001469 is the main entry for this sequence. A226158 is another version.
Cf. A005439 (Genocchi numbers of second kind).
Cf. A083007, A083008, A083009, A083010, A083011, A083012, A083013, A083014, A321595.

a := n -> n*euler(n-1,0); # Peter Luschny, Jul 13 2009

a[n_] := n*EulerE[n - 1, 0]; Table[a[n], {n, 1, 32}] (* Jean-François Alcover, Dec 08 2011, after Peter Luschny *)
Range[0, 31]! CoefficientList[ Series[ 2x/(1 + Exp[x]), {x, 0, 32}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Table[(-1)^n 2 n PolyLog[1 - n, -1], {n, 1, 32}] (* Peter Luschny, Aug 17 2021 *)

{a(n) = if( n<0, 0, n! * polcoeff( 2*x / (1 + exp(x + x * O(x^n))), n))}; /* Michael Somos, Jul 23 2005 */

/* From o.g.f. (Paul D. Hanna, Aug 03 2014): */
{a(n)=local(A=1); A=x*sum(m=0, n, m!*(-x)^m/(1-m*x)/prod(k=1,m,1 - k*x +x*O(x^n))); polcoeff(A, n)}
for(n=1, 32, print1(a(n), ", "))

from sympy import bernoulli
def A036968(n): return (2-(2<Chai Wah Wu, Apr 14 2023

# with a(1) = -1
[z*zeta(1-z)*(2^(z+1)-2) for z in (1..32)]  # Peter Luschny, Jun 28 2013

def A036968_list(len):
    e, f, R, C = 4, 1, [], [1]+[0]*(len-1)
    for n in (2..len-1):
        for k in range(n, 0, -1):
            C[k] = C[k-1] / (k+1)
        C[0] = -sum(C[k] for k in (1..n))
        R.append((2-e)*f*C[0])
        f *= n; e *= 2
    return R
print(A036968_list(34)) # Peter Luschny, Feb 22 2016

E.g.f.: 2*x/(exp(x)+1).
a(n) = 2*(1-2^n)*B_n (B = Bernoulli numbers). - Benoit Cloitre, Oct 26 2003
2*x/(exp(x)+1) = x + Sum_{n>=1} x^(2*n)*G_{2*n}/(2*n)!.
a(n) = Sum_{k=0..n-1} binomial(n,k) 2^k*B(k). - Peter Luschny, Apr 30 2009
From Sergei N. Gladkovskii, Dec 12 2012 to Nov 23 2013: (Start) Continued fractions:
E.g.f.: 2*x/(exp(x)+1) = x - x^2/2*G(0) where G(k) = 1 - x^2/(x^2 + 4*(2*k+1)*(2*k+3)/G(k+1)).
E.g.f.: 2/(E(0)+1) where E(k) = 1 + x/(2*k+1 - x*(2*k+1)/(x + (2*k+2)/E(k+1))).
G.f.: 2 - 1/G(0) where G(k) = 1 - x*(k+1)/(1 + x*(k+1)/(1 - x*(k+1)/(1 + x*(k+1)/G(k+1)))).
E.g.f.: 2*x/(1 + exp(x)) = 2*x-2 - 2*T(0), where T(k) = 4*k-1 + x/(2 - x/( 4*k+1 + x/(2 - x/T(k+1)))).
G.f.: 2 - Q(0)/(1-x+x^2) where Q(k) = 1 - x^4*(k+1)^4/(x^4*(k+1)^4 - (1 - x + x^2 + 2*x^2*k*(k+1))*(1 - x + x^2 + 2*x^2*(k+1)*(k+2))/Q(k+1)). (End)
a(n) = n*zeta(1-n)*(2^(n+1)-2) for n > 1. - Peter Luschny, Jun 28 2013
O.g.f.: x*Sum_{n>=0} n! * (-x)^n / (1 - n*x) / Product_{k=1..n} (1 - k*x). - Paul D. Hanna, Aug 03 2014
Sum_{n>=1} 1/a(2*n) = A321595. - Amiram Eldar, May 07 2021
a(n) = (-1)^n*2*n*PolyLog(1 - n, -1). - Peter Luschny, Aug 17 2021

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

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).

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A036968 Genocchi numbers (of first kind): expansion of 2*x/(exp(x)+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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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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A083011 a(n) = Sum_{k=0..n-1} 7^k*B(k)*binomial(n,k) where B(k) is the k-th Bernoulli number.

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

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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).

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.