A083007 -id:A083007 - 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

A002111 Glaisher's G numbers.

Original entry on oeis.org

1, 5, 49, 809, 20317, 722813, 34607305, 2145998417, 167317266613, 16020403322021, 1848020950359841, 252778977216700025, 40453941942593304589, 7488583061542051450829, 1587688770629724715374457, 382218817191632327375004833

Offset: 1

N. J. A. Sloane

Related to the formula Sum_{k>0} sin(kx)/k^(2n+1)=(-1)^(n+1)/2*x^(2n+1)/(2n+1)! * Sum_{i=0..2n} (2Pi/x)^i*B(i)*C(2n+1,i). - Benoit Cloitre, May 01 2002

Named after the English mathematician and astronomer James Whitbread Lee Glaisher (1848-1928). - Amiram Eldar, Jun 16 2021

			G.f. = x + 5*x^2 + 49*x^3 + 809*x^4 + 20317*x^5 + 722813*x^6 + 34607305*x^7 + ...

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. 76.
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).

Seiichi Manyama, Table of n, a(n) for n = 1..254 (first 50 terms from T. D. Noe)
Shaun Cooper, Cubic elliptic functions, Res. Lett. Inf. Math. Sci., Vol. 5 (2003), pp. 23-59, see page 30.
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.
J. W. L. Glaisher, On a set of coefficients analogous to the Eulerian numbers, Proc. London Math. Soc., Vol. 31 (1899), pp. 216-235.
René Gy, Bernoulli-Stirling Numbers, Integers, Vol. 20, (2020), #A67.
J. C. P. Miller, Letter to N. J. A. Sloane, Mar 26 1971.
N. J. A. Sloane, Transforms.
Wikipedia, Bernoulli Polynomials.
Index entries for sequences related to Glaisher's numbers

Cf. A083007, A033470.

read transforms; t1 := (3/2)/(1+exp(x)+exp(-x)); series(t1,x,50): t2 := SERIESTOLISTMULT(t1); [seq(n*t2[n],n=1..nops(t5))];

s[n_] := CoefficientList[Series[(1/2)*(Sin[t/2]/Sin[3*(t/2)]), {t, 0, 32}], t][[n + 1]]*n!*(-1)^Floor[n/2]; a[n_] := (-1)^n*(6*n + 3)*s[2*n]; Table[a[n], {n, 1, 16}] (* Jean-François Alcover, Mar 22 2011, after Michael Somos' formula *)
a[ n_] := If[ n < 1, 0, (2 n + 1)! SeriesCoefficient[ 3 / (2 + 4 Cos[x]), {x, 0, 2 n}]]; (* Michael Somos, Jun 01 2012 *)

{a(n) = if( n<1, 0, n*=2; (n+1)! * polcoeff( 3 / (2 + 4 * cos( x + O(x^n))), n))}; /* Michael Somos, Feb 26 2004 */

a(n)=if(n<1,0,-(-1)^n*sum(i=0,2*n,binomial(2*n+1,i)*bernfrac(i)*3^i)) \\ Benoit Cloitre, May 01 2002

def A002111(n):
    return add(add(add(((-1)^(n+1-v)/(j+1))*binomial(2*n+1,k)*binomial(j,v)*(3*v)^k for v in (0..j)) for j in (0..k)) for k in (0..2*n+1))
[A002111(n) for n in (1..16)]  # Peter Luschny, Jun 03 2013

To get these numbers, expand the e.g.f. (3/2)/(1+exp(x)+exp(-x)), multiply coefficient of x^n by (n+1)! and take absolute values.
Or expand the e.g.f. (3/2)/(1+2*cos(x)) and multiply coefficient of x^n by (n+1)!. - Herb Conn, Feb 25 2002
a(n) = (2n+1)*I(n), where I(n) is given by A047788/A047789.
a(n) = Sum_{i=0, 2n} B(i)*C(2n+1, i)*3^i where B(i) are the Bernoulli numbers, C(2n, i) the binomial numbers. - Benoit Cloitre, May 01 2002
a(n) = (-1)^n * (6*n + 3) * s(2*n), if n>0, where s(n) are the cubic Bernoulli numbers. - Michael Somos, Feb 26 2004
E.g.f.: 3*x / (2 + 4*cos(x)) = Sum_{n>=0} a(n) * x^(2*n+1) / (2*n+1)!. - Michael Somos, Feb 26 2004
E.g.f.: E(x) = (3/2)/(1+2*cos(x)) - 1/2 = x^2/(3*G(0)+x^2); G(k) = 2*(2*k+1)*(k+1) - x^2 + 2*x^2*(2*k+1)*(k+1)/G(k+1); (continued fraction Euler's kind, 1-step). Let f[n]:=coeftayl(E(x), x=0, n) then: A002111[n]=f[2*n+2]*((2*n+3)!). - Sergei N. Gladkovskii, Jan 14 2012
a(n) = Sum_{k=0..2n+1} Sum_{j=0..k} Sum_{v=0..j} ((-1)^(n-v+1)/(j+1))* binomial(2*n+1,k)*binomial(j,v)*(3*v)^k. - Peter Luschny, Jun 03 2013
a(n) ~ (2*n+1)! * sqrt(3) * (3/(2*Pi))^(2*n+1). - Vaclav Kotesovec, Jul 30 2013
From Peter Bala, Mar 02 2015: (Start)
a(n) = (-1)^(n+1)*3^(2*n+1)*B(2*n+1,1/3), where B(n,x) denotes the n-th Bernoulli polynomial. Cf. A009843, A069852, A069994.
Conjecturally, a(n) = the unsigned numerator of B(2*n+1,1/3). Cf. A033470.
Essentially a bisection of |A083007|.
G.f. for signed version of sequence: 1/2 + 1/2*Sum_{n >= 0} { 1/(n+1) * Sum_{k = 0..n} (-1)^(k+1)*binomial(n,k)/( (1 - (3*k + 1)*x)*(1 - (3*k + 2)*x) ) } = x^2 - 5*x^4 + 49*x^6 - .... (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).

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

Original entry on oeis.org

0, 1, -6, 15, 36, -335, -1098, 16955, 73032, -1503963, -8075430, 204957775, 1319806188, -39666688711, -297958666242, 10337889346275, 88743928066704, -3489994294713779, -33703905982634334, 1481439997178305655, 15896303102840841780, -772269573963075710367

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..441

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083010, A083012, A083013, A083014.

Range[0, 15]! CoefficientList[ Series[ 7x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x] + Exp[ 6x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)

a(n)=sum(k=0,n-1,7^k*binomial(n,k)*bernfrac(k))

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

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

Original entry on oeis.org

0, 1, -7, 21, 49, -595, -1911, 39109, 165473, -4525731, -23883335, 805349237, 5097585297, -203564524787, -1503073984279, 69292329479205, 584713994953921, -30553447357629763, -290046835163027943, 16939595863125337813, 178676615255242261745

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..432

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083010, A083011, A083013, A083014.

Range[0, 15]! CoefficientList[ Series[ 8x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x] + Exp[ 6x] + Exp[ 7x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)

a(n)=sum(k=0,n-1,8^k*binomial(n,k)*bernfrac(k))

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

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

Original entry on oeis.org

0, 1, -8, 28, 64, -980, -3104, 81172, 339328, -11878244, -61958240, 2674671076, 16735235392, -855605816468, -6245150369696, 368601472639540, 3074742020313856, -205700802920736452, -1930357641628367072, 144338957346266943364, 1505019970814899568320

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..424

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083010, A083011, A083012, A083014.

Range[0, 15]! CoefficientList[ Series[ 9x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x] + Exp[ 6x] + Exp[ 7x] + Exp[ 8x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Table[Sum[9^k BernoulliB[k]Binomial[n,k],{k,0,n-1}],{n,0,20}] (* Harvey P. Dale, Apr 13 2016 *)

a(n)=sum(k=0,n-1,9^k*binomial(n,k)*bernfrac(k))

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

A083014 a(n) = Sum_{k=0..n-1} 10^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.

Original entry on oeis.org

0, 1, -9, 36, 81, -1524, -4779, 155316, 643761, -28041444, -145069299, 7794224196, 48371836041, -3078058903764, -22284938832219, 1637087002046676, 13545357290061921, -1127884947406124484, -10498665795419017539, 977073296798704710756

Offset: 0

Benoit Cloitre, May 31 2003

sign

Seiichi Manyama, Table of n, a(n) for n = 0..417
Ira M. Gessel, On the Almkvist-Meurman Theorem for Bernoulli Polynomials, Integers (2023) Vol. 23, #A14.

Cf. A001469.

Cf. A036968, A083007, A083008, A083009, A083010, A083011, A083012, A083013.

Range[0, 15]! CoefficientList[ Series[ 10x/(1 + Exp[x] + Exp[ 2x] + Exp[ 3x] + Exp[ 4x] + Exp[ 5x] + Exp[ 6x] + Exp[ 7x] + Exp[ 8x] + Exp[ 9x]), {x, 0, 15}], x] (* Robert G. Wilson v, Oct 26 2012 *)
Array[Sum[10^k*BernoulliB[k]*Binomial[#, k], {k, 0, # - 1}] &, 20, 0] (* Michael De Vlieger, Feb 14 2023 *)

a(n)=sum(k=0,n-1,10^k*binomial(n,k)*bernfrac(k))

E.g.f.: 10*x/(Sum_{i=0..9} exp(i*x)). - Alois P. Heinz, Sep 28 2016

Offset changed to 0 by Seiichi Manyama, Sep 28 2016

A239977 a(n) = -Sum_{k=0..n} binomial(n, k)*A226158(k).

Original entry on oeis.org

0, 1, 3, 6, 9, 10, 9, 14, 33, 18, -135, 22, 2097, 26, -38199, 30, 929601, 34, -28820583, 38, 1109652945, 42, -51943281687, 46, 2905151042529, 50, -191329672483911, 54, 14655626154768753, 58, -1291885088448017655, 62, 129848163681107302017, 66

Offset: 0

Paul Curtz, Mar 30 2014

Let T(n, k) denote the difference table of a(n).
(-1)^(k+1)*T(3, 3+k) = T(k+3, 3) for k >= 0.
Without the first two rows and the first two columns we have the core of the Genocchi numbers, like A240581(n)/A239315(n) for the Bernoulli numbers. See A226158(n).
   0,   1,   3,   6,   9,  10, ...
   1,   2,   3,   3,   1,  -1, ...
   1,   1,   0,  -2,  -2,   6, ...
   0,  -1,  -2,   0,   8,   8, ...
  -1,  -1,   2,   8,   0, -56, ...
   0,   3,   6,  -8, -56,   0, ...
The definition reflects the identity 2*((1-2^n)*B(n,1) + n) =
2*Sum_{k=0..n} C(n,k)*(2^k-1)*B(k,1) where B(n,x) denotes the Bernoulli polynomials. - Peter Luschny, Apr 16 2014

B. Sury, The value of Bernoulli Polynomials at rational numbers, Bull. London Math. Soc. 25 (1993), 327-29.

Cf. A083007.

[0,1] cat [2*((1 - 2^n)*Bernoulli(n) + n): n in [2..40]]; // Vincenzo Librandi, Mar 03 2015

A239977 := n -> 2*((1-2^n)*bernoulli(n,1) + n):
seq(A239977(n), n=0..33); # Peter Luschny, Mar 08 2015

a[n_] := (EulerE[n-1, 0]+2)*n; a[0] = 0; a[1] = 1; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Apr 02 2014 *)

a(n) = 2*n + A226158(n).
a(2n) is divisible by 3.
a(2n+1) = A133653(n).
a(n) = 2*((1 - 2^n)*B(n, 1) + n), B(n, x) the Bernoulli polynomial. - Peter Luschny, Apr 16 2014
From Peter Bala, Mar 02 2015: (Start)
a(n) = (-2)^n * ( B(n,-1/2) - B(n,0) ), where B(n,x) denotes the n-th Bernoulli polynomial. More generally, for any nonzero integer k, k^n*( B(n,1/k) - B(n,0) ) is an integer for n >= 0. Cf. A083007.
a(0) = 0 and for n >= 1, a(n) = 1 - 1/(n + 1)*Sum_{k = 1..n-1} (-2)^(n-k)*binomial(n+1,k)*a(k).
E.g.f.: 2*x*exp(2*x)/(1 + exp(x)) = x + 3*x^2/2! + 6*x^3/3! + .... (End)
a(n) = Sum_{k=0..n-1} 2^k*binomial(n, k)*Bernoulli(k, 1). - Peter Luschny, Aug 17 2021

cp's OEIS Frontend

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

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A002111 Glaisher's G numbers.

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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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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.

A083014 a(n) = Sum_{k=0..n-1} 10^kB(k)binomial(n,k) where B(k) is the k-th Bernoulli number.