A130189 -id:A130189 - cp's OEIS frontend

A027641 Numerator of Bernoulli number B_n.

1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, -174611, 0, 854513, 0, -236364091, 0, 8553103, 0, -23749461029, 0, 8615841276005, 0, -7709321041217, 0, 2577687858367, 0, -26315271553053477373, 0, 2929993913841559, 0, -261082718496449122051

Offset: 0

N. J. A. Sloane

a(n)/A027642(n) (Bernoulli numbers) provide the a-sequence for the Sheffer matrix A094816 (coefficients of orthogonal Poisson-Charlier polynomials). See the W. Lang link under A006232 for a- and z-sequences for Sheffer matrices. The corresponding z-sequence is given by the rationals A130189(n)/A130190(n).
Harvey (2008) describes a new algorithm for computing Bernoulli numbers. His method is to compute B(k) modulo p for many small primes p and then reconstruct B(k) via the Chinese Remainder Theorem. The time complexity is O(k^2 log(k)^(2+eps)). The algorithm is especially well-suited to parallelization. - Jonathan Vos Post, Jul 09 2008
Regard the Bernoulli numbers as forming a vector = B_n, and the variant starting (1, 1/2, 1/6, 0, -1/30, ...), (i.e., the first 1/2 has sign +) as forming a vector Bv_n. The relationship between the Pascal triangle matrix, B_n, and Bv_n is as follows: The binomial transform of B_n = Bv_n. B_n is unchanged when multiplied by the Pascal matrix with rows signed (+-+-, ...), i.e., (1; -1,-1; 1,2,1; ...). Bv_n is unchanged when multiplied by the Pascal matrix with columns signed (+-+-, ...), i.e., (1; 1,-1; 1,-2,1; 1,-3,3,-1; ...). - Gary W. Adamson, Jun 29 2012
The sequence of the Bernoulli numbers B_n = a(n)/A027642(n) is the inverse binomial transform of the sequence {A164555(n)/A027642(n)}, illustrated by the fact that they appear as top row and left column in A190339. - Paul Curtz, May 13 2016
Named by de Moivre (1773; "the numbers of Mr. James Bernoulli") after the Swiss mathematician Jacob Bernoulli (1655-1705). - Amiram Eldar, Oct 02 2023

			B_n sequence begins 1, -1/2, 1/6, 0, -1/30, 0, 1/42, 0, -1/30, 0, 5/66, 0, -691/2730, 0, 7/6, 0, -3617/510, ...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 810.
Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 49.
Harold T. Davis, Tables of the Mathematical Functions. Vols. 1 and 2, 2nd ed., 1963, Vol. 3 (with V. J. Fisher), 1962; Principia Press of Trinity Univ., San Antonio, TX, Vol. 2, p. 230.
Harold M. Edwards, Riemann's Zeta Function, Academic Press, NY, 1974; see p. 11.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, Section 1.6.1.
Herman H. Goldstine, A History of Numerical Analysis, Springer-Verlag, 1977; Section 2.6.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 137.
Hans Rademacher, Topics in Analytic Number Theory, Springer, 1973, Chap. 1.

T. D. Noe, Table of n, a(n) for n = 0..200
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
C. M. Bender and K. A. Milton, Continued fraction as a discrete nonlinear transform, arXiv:hep-th/9304052, 1993.
Beáta Bényi and Péter Hajnal, Poly-Bernoulli Numbers and Eulerian Numbers, arXiv:1804.01868 [math.CO], 2018.
H. Bergmann, Eine explizite Darstellung der Bernoullischen Zahlen, Math. Nach. 34 (1967), 377-378. Math Rev 36#4030.
Richard P. Brent and David Harvey, Fast computation of Bernoulli, Tangent and Secant numbers, arXiv preprint arXiv:1108.0286 [math.CO], 2011.
K.-W. Chen, Algorithms for Bernoulli numbers and Euler numbers, J. Integer Sequences, 4 (2001), #01.1.6.
W. Y.C. Chen, J. J. F. Guo and L. X. W. Wang, Log-behavior of the Bernoulli Numbers, arXiv:1208.5213 [math.CO], 2012-2013.
Abraham de Moivre, The Doctrine of Chances, 3rd edition, London, 1733, p. 95.
K. Dilcher, A Bibliography of Bernoulli Numbers (Alphabetically Indexed Authorwise).
Bakir Farhi, Formulas Involving Bernoulli and Stirling Numbers of Both Kinds, Journal of Integer Sequences, Vol. 28 (2025), Article 25.2.6. See p. 16.
Ghislain R. Franssens, On a Number Pyramid Related to the Binomial, Deleham, Eulerian, MacMahon and Stirling number triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.4.1.
H. W. Gould and J. Quaintance, Bernoulli Numbers and a New Binomial Transform Identity, J. Int. Seq. 17 (2014), Article 14.2.2.
M.-P. Grosset and A. P. Veselov, Bernoulli numbers and solitons, arXiv:math/0503175 [math.GM], 2005.
David Harvey, A multimodular algorithm for computing Bernoulli numbers, arXiv:0807.1347 [math.NT], Jul 08 2008.
A. Iványi, Leader election in synchronous networks, Acta Univ. Sapientiae, Mathematica, 5, 2 (2013) 54-82.
Sumit Kumar Jha, A new explicit formula for the Bernoulli numbers in terms of the Stirling numbers of the second kind, Notes on Number Theory and Discrete Mathematics (2020) Vol. 26, No. 2, 148-151.
M. Kaneko, The Akiyama-Tanigawa algorithm for Bernoulli numbers, J. Integer Sequences, 3 (2000), Article 00.2.9.
Wolfdieter Lang, On Sums of Powers of Arithmetic Progressions, and Generalized Stirling, Eulerian and Bernoulli numbers, arXiv:1707.04451 [math.NT], 2017.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014), Article 14.4.6.
F. Luca and P. Stanica, On some conjectures on the monotonicity of some arithmetical sequences, J. Combin. Number Theory 4 (2012) 1-10.
Peter Luschny, The Bernoulli Manifesto. A survey on the occasion of the 300th anniversary of the publication of Jacob Bernoulli's Ars Conjectandi, 1713-2013.
Romeo Meštrović, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), Article 14.8.4.
Hisanori Mishima, Bernoulli numbers (n = 2 to 114), (n = 116 to 154) (Factorizations).
Ben Moonen, A remark on the paper of Deninger and Murre, arXiv:2407.05837 [math.AG], 2024. See p. 6.
A. F. Neto, Carlitz's Identity for the Bernoulli Numbers and Zeon Algebra, J. Int. Seq. 18 (2015), Article 15.5.6.
A. F. Neto, A note of a Theorem of Guo, Mezo, and Qi, J. Int. Seq. 19 (2016) Article 16.4.8.
Niels Nielsen, Traite Elementaire des Nombres de Bernoulli, Gauthier-Villars, 1923, pp. 398.
Simon Plouffe, The First 498 Bernoulli numbers. [Project Gutenberg Etext]
Ed Sandifer, How Euler Did It, Bernoulli numbers.
N. J. A. Sloane, "A Handbook of Integer Sequences" Fifty Years Later, arXiv:2301.03149 [math.NT], 2023, p. 5.
J. Singh, On an Arithmetic Convolution, J. Int. Seq. 17 (2014) Article 14.6.7.
J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 5.
Eric Weisstein's World of Mathematics, Bernoulli Number.
Eric Weisstein's World of Mathematics, Polygamma Function.
Roman Witula, Damian Slota and Edyta Hetmaniok, Bridges between different known integer sequences, Annales Mathematicae et Informaticae, 41 (2013) pp. 255-263.
Wolfram Research, Generating functions of B_n & B_2n.
Index entries for "core" sequences.
Index entries for sequences related to Bernoulli numbers.

This is the main entry for the Bernoulli numbers and has all the references, links and formulas. Sequences A027642 (the denominators of B_n) and A000367/A002445 = B_{2n} are also important!
A refinement is A194587.
Cf. A000146, A002882, A003245, A127187, A127188, A074909, A164555, A176327, A190339, A131689.
Cf. A094816, A130189, A130190, A155585.

[Numerator(Bernoulli(n)): n in [0..40]]; // Vincenzo Librandi, Mar 17 2014

B := n -> add((-1)^m*m!*Stirling2(n, m)/(m+1), m=0..n);
B := n -> bernoulli(n);
seq(numer(bernoulli(n)), n=0..40); # Zerinvary Lajos, Apr 08 2009

Table[ Numerator[ BernoulliB[ n]], {n, 0, 40}] (* Robert G. Wilson v, Oct 11 2004 *)
Numerator[ Range[0, 40]! CoefficientList[ Series[x/(E^x - 1), {x, 0, 40}], x]]
Numerator[CoefficientList[Series[PolyGamma[1, 1/x]/x - x, {x, 0, 40}, Assumptions -> x > 0], x]] (* Vladimir Reshetnikov, Apr 24 2013 *)

B(n):=(-1)^((n))*sum((stirling1(n,k)*stirling2(n+k,n))/binomial(n+k,k),k,0,n);
makelist(num(B(n)),n,0,20); /* Vladimir Kruchinin, Mar 16 2013 */

PARI
```
a(n)=numerator(bernfrac(n))
```

from sympy import bernoulli
from fractions import Fraction
[bernoulli(i).as_numer_denom()[0] for i in range(51)]  # Indranil Ghosh, Mar 18 2017

from sympy import bernoulli
def A027641(n): return bernoulli(n).p
print([A027641(n) for n in range(80)])  # M. F. Hasler, Jun 11 2019

[bernoulli(n).numerator() for n in range(41)]  # Peter Luschny, Feb 19 2016

# Alternatively:
def A027641_list(len):
    f, R, C = 1, [1], [1]+[0]*(len-1)
    for n in (1..len-1):
        f *= n
        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((C[0]*f).numerator())
    return R
A027641_list(41)  # Peter Luschny, Feb 20 2016

E.g.f: x/(exp(x) - 1); take numerators.
Recurrence: B^n = (1+B)^n, n >= 2 (interpreting B^j as B_j).
B_{2n}/(2n)! = 2*(-1)^(n-1)*(2*Pi)^(-2n) Sum_{k>=1} 1/k^(2n) (gives asymptotics) - Rademacher, p. 16, Eq. (9.1). In particular, B_{2*n} ~ (-1)^(n-1)*2*(2*n)!/(2*Pi)^(2*n).
Sum_{i=1..n-1} i^k = ((n+B)^(k+1)-B^(k+1))/(k+1) (interpreting B^j as B_j).
B_{n-1} = - Sum_{r=1..n} (-1)^r binomial(n, r) r^(-1) Sum_{k=1..r} k^(n-1). More concisely, B_n = 1 - (1-C)^(n+1), where C^r is replaced by the arithmetic mean of the first r n-th powers of natural numbers in the expansion of the right-hand side. [Bergmann]
Sum_{i>=1} 1/i^(2k) = zeta(2k) = (2*Pi)^(2k)*|B_{2k}|/(2*(2k)!).
B_{2n} = (-1)^(m-1)/2^(2m+1) * Integral{-inf..inf, [d^(m-1)/dx^(m-1) sech(x)^2 ]^2 dx} (see Grosset/Veselov).
Let B(s,z) = -2^(1-s)(i/Pi)^s s! PolyLog(s,exp(-2*i*Pi/z)). Then B(2n,1) = B_{2n} for n >= 1. Similarly the numbers B(2n+1,1), which might be called Co-Bernoulli numbers, can be considered, and it is remarkable that Leonhard Euler in 1755 already calculated B(3,1) and B(5,1) (Opera Omnia, Ser. 1, Vol. 10, p. 351). (Cf. the Luschny reference for a discussion.) - Peter Luschny, May 02 2009
The B_n sequence is the left column of the inverse of triangle A074909, the "beheaded" Pascal's triangle. - Gary W. Adamson, Mar 05 2012
From Sergei N. Gladkovskii, Dec 04 2012: (Start)
E.g.f. E(x)= 2 - x/(tan(x) + sec(x) - 1)= Sum_{n>=0} a(n)*x^n/n!, a(n)=|B(n)|, where B(n) is Bernoulli number B_n.
E(x)= 2 + x - B(0), where B(k)=  4*k+1 + x/(2 + x/(4*k+3 - x/(2 - x/B(k+1)))); (continued fraction, 4-step). (End)
E.g.f.: x/(exp(x)-1)= U(0); U(k)= 2*k+1 - x(2*k+1)/(x + (2*k+2)/(1 + x/U(k+1))); (continued fraction). - Sergei N. Gladkovskii, Dec 05 2012
E.g.f.: 2*(x-1)/(x*Q(0)-2) where Q(k) = 1 + 2*x*(k+1)/((2*k+1)*(2*k+3) - x*(2*k+1)*(2*k+3)^2/(x*(2*k+3) + 4*(k+1)*(k+2)/Q(k+1))); (recursively defined continued fraction). - Sergei N. Gladkovskii, Feb 26 2013
a(n) = numerator(B(n)), B(n) = (-1)^n*Sum_{k=0..n} Stirling1(n,k) * Stirling2(n+k,n) / binomial(n+k,k). - Vladimir Kruchinin, Mar 16 2013
E.g.f.: x/(exp(x)-1) = E(0) where E(k) = 2*k+1 - x/(2 + x/E(k+1)); (continued fraction). - Sergei N. Gladkovskii, Mar 16 2013
G.f. for Bernoulli(n) = a(n)/A027642(n): psi_1(1/x)/x - x, where psi_n(z) is the polygamma function, psi_n(z) = (d/dz)^(n+1) log(Gamma(z)). - Vladimir Reshetnikov, Apr 24 2013
E.g.f.: 2*E(0) - 2*x, where E(k)= x + (k+1)/(1 + 1/(1 - x/E(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Jul 10 2013
B_n = Sum_{m=0..n} (-1)^m *A131689(n, m)/(m + 1), n >= 0. See one of the Maple programs. - Wolfdieter Lang, May 05 2017
a(n) = numerator((-1)^n*A155585(n-1)*n/(4^n-2^n)), for n>=1. - Mats Granvik, Nov 26 2017
From Artur Jasinski, Dec 30 2020: (Start)
a(n) = numerator(-2*cos(Pi*n/2)*Gamma(n+1)*zeta(n)/(2*Pi)^n), for n=0 and n>1.
a(n) = numerator(-n*zeta(1-n)), for n=0 and n>1. (End)
a(n) = numerator(Sum_{k=0..n-1} (-1)^(k-1)*k!*Stirling2(n-1,k) / ((k+1)*(k+2))), for n>0 (see Jha link). - Bill McEachen, Jul 17 2025

A094816 Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 8, 6, 1, 1, 24, 29, 10, 1, 1, 89, 145, 75, 15, 1, 1, 415, 814, 545, 160, 21, 1, 1, 2372, 5243, 4179, 1575, 301, 28, 1, 1, 16072, 38618, 34860, 15659, 3836, 518, 36, 1, 1, 125673, 321690, 318926, 163191, 47775, 8274, 834, 45, 1, 1, 1112083, 2995011

Offset: 0

Philippe Deléham, Jun 12 2004

The a-sequence for this Sheffer matrix is A027641(n)/A027642(n) (Bernoulli numbers) and the z-sequence is A130189(n)/ A130190(n). See the W. Lang link.
Take the lower triangular matrix in A049020 and invert it, then read by rows! - N. J. A. Sloane, Feb 07 2009
Exponential Riordan array [exp(x), log(1/(1-x))]. Equal to A007318*A132393. - Paul Barry, Apr 23 2009
A signed version of the triangle appears in [Gessel]. - Peter Bala, Aug 31 2012
T(n,k) is the number of permutations over all subsets of {1,2,...,n} (Cf. A000522) that have exactly k cycles.  T(3,2) = 6: We permute the elements of the subsets {1,2}, {1,3}, {2,3}. Each has one permutation with 2 cycles.  We permute the elements of {1,2,3} and there are three permutations that have 2 cycles. 3*1 + 1*3 = 6. - Geoffrey Critzer, Feb 24 2013
From Wolfdieter Lang, Jul 28 2017: (Start)
In Chihara's book the row polynomials (with rising powers) are the Charlier polynomials (-1)^n*C^(a)_n(-x), with a = -1, n >= 0. See  p. 170, eq. (1.4).
In Ismail's book the present Charlier polynomials are denoted by C_n(-x;a=1) on p. 177, eq. (6.1.25). (End)
The triangle T(n,k) is a representative of the parametric family of triangles T(m,n,k), whose columns are the coefficients of the standard expansion of the function f(x) = (-log(1-x))^(k)*exp(-m*x)/k! for the case m=-1. See A381082. - Igor Victorovich Statsenko, Feb 14 2025

			From _Paul Barry_, Apr 23 2009: (Start)
Triangle begins
  1;
  1,     1;
  1,     3,     1;
  1,     8,     6,     1;
  1,    24,    29,    10,     1;
  1,    89,   145,    75,    15,    1;
  1,   415,   814,   545,   160,   21,   1;
  1,  2372,  5243,  4179,  1575,  301,  28,  1;
  1, 16072, 38618, 34860, 15659, 3836, 518, 36, 1;
Production matrix is
  1, 1;
  0, 2, 1;
  0, 1, 3,  1;
  0, 1, 3,  4,  1;
  0, 1, 4,  6,  5,  1;
  0, 1, 5, 10, 10,  6,  1;
  0, 1, 6, 15, 20, 15,  7,  1;
  0, 1, 7, 21, 35, 35, 21,  8, 1;
  0, 1, 8, 28, 56, 70, 56, 28, 9, 1; (End)

T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon and Breach, New York, London, Paris, 1978, Ch. VI, 1., pp. 170-172.
Classical and Quantum Orthogonal Polynomials in One Variable, Cambridge University Press, 2005, EMA, Vol. 98, p. 177.

Paul Barry, The Restricted Toda Chain, Exponential Riordan Arrays, and Hankel Transforms, J. Int. Seq. 13 (2010) # 10.8.4, example 6.
Paul Barry, Exponential Riordan Arrays and Permutation Enumeration, J. Int. Seq. 13 (2010) # 10.9.1, example 8.
Paul Barry, Riordan Arrays, Orthogonal Polynomials as Moments, and Hankel Transforms, J. Int. Seq. 14 (2011) # 11.2.2, example 22.
Paul Barry, Combinatorial polynomials as moments, Hankel transforms and exponential Riordan arrays, arXiv preprint arXiv:1105.3044 [math.CO], 2011, also J. Int. Seq. 14 (2011) 11.6.7.
Ira Gessel, Congruences for Bell and Tangent numbers, The Fibonacci Quarterly, Vol. 19, Number 2, 1981.
Aoife Hennessy and Paul Barry, Generalized Stirling Numbers, Exponential Riordan Arrays, and Orthogonal Polynomials, J. Int. Seq. 14 (2011) # 11.8.2.
Marin Knežević, Vedran Krčadinac, and Lucija Relić, Matrix products of binomial coefficients and unsigned Stirling numbers, arXiv:2012.15307 [math.CO], 2020.
Wolfdieter Lang, First 10 rows and more.
Wolfdieter Lang, On Generating functions of Diagonals Sequences of Sheffer and Riordan Number Triangles, arXiv:1708.01421 [math.NT], August 2017.
W. F. Lunnon, P. A. B. Pleasants, and N. M. Stephens, Arithmetic properties of Bell numbers to a composite modulus I, Acta Arithmetica 35 (1979), pp. 1-16.

Columns k=0..4 give A000012, A002104, A381021, A381022, A381023.
Diagonals: A000012, A000217.
Row sums A000522, alternating row sums A024000.
KummerU(-n,1-n-x,z): this sequence (z=1), |A137346| (z=2), A327997 (z=3).

A094816 := (n,k) -> (-1)^(n-k)*add(binomial(-j-1,-n-1)*Stirling1(j,k), j=0..n):
seq(seq(A094816(n, k), k=0..n), n=0..9); # Peter Luschny, Apr 10 2016

nn=10;f[list_]:=Select[list,#>0&];Map[f,Range[0,nn]!CoefficientList[Series[ Exp[x]/(1-x)^y,{x,0,nn}],{x,y}]]//Grid  (* Geoffrey Critzer, Feb 24 2013 *)
Flatten[Table[(-1)^(n-k) Sum[Binomial[-j-1,-n-1] StirlingS1[j,k],{j,0,n}], {n,0,9},{k,0,n}]] (* Peter Luschny, Apr 10 2016 *)
p[n_] := HypergeometricU[-n, 1 - n - x, 1];
Table[CoefficientList[p[n], x], {n,0,9}] // Flatten (* Peter Luschny, Oct 27 2019 *)

{T(n, k)= local(A); if( k<0 || k>n, 0, A = x * O(x^n); polcoeff( n! * polcoeff( exp(x + A) / (1 - x + A)^y, n), k))} /* Michael Somos, Nov 19 2006 */

def a_row(n):
    s = sum(binomial(n,k)*rising_factorial(x,k) for k in (0..n))
    return expand(s).list()
[a_row(n) for n in (0..9)] # Peter Luschny, Jun 28 2019

E.g.f.: exp(t)/(1-t)^x = Sum_{n>=0} C(x,n)*t^n/n!.
Sum_{k = 0..n} T(n, k)*x^k = C(x, n), Charlier polynomials; C(x, n)= A024000(n), A000012(n), A000522(n), A001339(n), A082030(n), A095000(n), A095177(n), A096307(n), A096341(n), A095722(n), A095740(n) for x = -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 respectively. - Philippe Deléham, Feb 27 2013
T(n+1, k) = (n+1)*T(n, k) + T(n, k-1) - n*T(n-1, k) with T(0, 0) = 1, T(0, k) = 0 if k>0, T(n, k) = 0 if k<0.
PS*A008275*PS as infinite lower triangular matrices, where PS is a triangle with PS(n, k) = (-1)^k*A007318(n, k). PS = 1/PS. - Gerald McGarvey, Aug 20 2009
T(n,k) = (-1)^(n-k)*Sum_{j=0..n} C(-j-1, -n-1)*S1(j, k) where S1 are the signed Stirling numbers of the first kind. - Peter Luschny, Apr 10 2016
Absolute values T(n,k) of triangle (-1)^(n+k) T(n,k) where row n gives coefficients of x^k, 0 <= k <= n, in expansion of Sum_{k=0..n} binomial(n,k) (-1)^(n-k) x^{(k)}, where x^{(k)} := Product_{i=0..k-1} (x-i), k >= 1, and x^{(0)} := 1, the falling factorial powers. - Daniel Forgues, Oct 13 2019
From Peter Bala, Oct 23 2019: (Start)
The n-th row polynomial is
   R(n, x) = Sum_{k = 0..n} (-1)^k*binomial(n, k)*k! * binomial(-x, k).
These polynomials occur in series acceleration formulas for the constant
   1/e = n! * Sum_{k >= 0} (-1)^k/(k!*R(n,k)*R(n,k+1)), n >= 0. (cf. A068985, A094816 and A137346). (End)
R(n, x) = KummerU[-n, 1 - n - x, 1]. - Peter Luschny, Oct 27 2019
Sum_{j=0..m} (-1)^(m-j) * Bell(n+j) * T(m,j) = m! * Sum_{k=0..n} binomial(k,m) * Stirling2(n,k). - Vaclav Kotesovec, Aug 06 2021
From Natalia L. Skirrow, Jun 11 2025: (Start)
G.f.: 2F0(1,y;x/(1-x)) / (1-x).
Polynomial for the n-th row is R(n,y) = 2F0(-n,y;-1).
Falling g.f. for n-th row: Sum_{k=0..n} a(n,k)*(y)_k = [x^0] 2F0(1,-n;-1/x) * (1+log(1/(1-x)))^y = [x^n] e^x * Gamma(n+1,x) * (1+log(1/(1-x)))^y, where (y)_k = y!/(y-k)! denotes the falling factorial. (End)

A130190 Denominators of z-sequence for the Sheffer matrix (triangle) A094816 (coefficients of Poisson-Charlier polynomials).

Original entry on oeis.org

1, 2, 6, 4, 15, 12, 42, 24, 90, 10, 33, 8, 910, 105, 90, 48, 255, 180, 3990, 420, 6930, 330, 345, 720, 13650, 273, 378, 28, 145, 20, 14322, 2464, 117810, 3570, 7, 24, 1919190, 1729, 2730, 840, 9471, 13860, 99330, 1540, 217350, 4830, 4935, 10080, 324870

Offset: 0

Wolfdieter Lang, Jun 01 2007

The numerators are given in A130189.
See A130189 for the W. Lang link on z-sequences for Sheffer matrices.
The prime factors of each a(n) are such that n!/a(n) has the prime, p = n+1, as the denominator of its reduced fraction, and if n+1 is not prime then n!/a(n) is an integer, except at n = 3, which has denominator = 2. Also see alternate formula for a(n) below. - Richard R. Forberg, Dec 28 2014
As implied above, at n = p-1 the largest prime factor of a(n) is p. For a(m), where m is an integer within the set given by A089965, the two largest prime factors of a(m) are m+1 and (m+1)/2. Furthermore, it appears, when n+1 is not a prime no prime factor of a(n) is greater than k/2, where k is the next higher value of n where n+1 is prime. Two examples at this upper limit of k/2 are n = 104 and 105, where the highest prime factor of a(n) is 53; it is then at n = k = 106 where n+1 is prime. - Richard R. Forberg, Jan 01 2015

G. C. Greubel, Table of n, a(n) for n = 0..1000

Cf. A089965, A094816, A248716, A130189.

seq(denom(add(Stirling2(n,k)/(k+1),k=0..n)),n=0..20); # Peter Luschny, Apr 28 2009

Denominator[Table[(1/Exp[1])* Sum[Sum[j^n/k!, {j, 0, k}], {k, 0, Infinity}], {n, 0, 100}]] (* Richard R. Forberg, Dec 28 2014 *)
Table[Denominator[Sum[StirlingS2[n, k]/(k + 1), {k, 0, n}]], {n, 0, 50}] (* G. C. Greubel, Jul 10 2018 *)

a(n) = denominator(sum(k=0, n, stirling(n, k, 2)/(k+1))); \\ Michel Marcus, Jan 15 2015, after Maple

a(n) = denominator(z(n)),n>=0, with the e.g.f. for z(n) given in A130189.
Denominator of Sum_{k=0..n} A048993(n,k)/(k+1).  - Peter Luschny, Apr 28 2009
Alternate: a(n) = denominator((1/e)*Sum_{k>=0}*(Sum_{j=0..k} j^n/k!)). NOTE: Numerators are different from A130189, and given by A248716. - Richard R. Forberg, Dec 28 2014
This more generalized expression ((1/e)*Sum_{k>=0} (Sum_{j=0..k} (j+m)^n/k!)), gives the same denominators for any integer m. - Richard R. Forberg, Jan 14 2015

cp's OEIS Frontend

A027641 Numerator of Bernoulli number B_n.

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A094816 Triangle read by rows: T(n,k) are the coefficients of Charlier polynomials: A046716 transposed, for 0 <= k <= n.

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A130190 Denominators of z-sequence for the Sheffer matrix (triangle) A094816 (coefficients of Poisson-Charlier polynomials).

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