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

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

Author

Wolfdieter Lang, Jun 01 2007

Keywords

• nonn
• easy

Comments

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

Links

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

Crossrefs

Cf. A089965, A094816, A248716, A130189.

Programs

Maple

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

Mathematica

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 *)

PARI

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

Formula

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