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A130190 Denominators of z-sequence for the Sheffer matrix (triangle) A094816 (coefficients of Poisson-Charlier polynomials). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

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

Table of n, a(n) for n=0..48.

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

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 *)

PROG

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

CROSSREFS

Cf. A089965, A094816, A248716, A130189.

Sequence in context: A111807 A069914 A200746 * A110346 A095754 A226718

Adjacent sequences:  A130187 A130188 A130189 * A130191 A130192 A130193

KEYWORD

nonn,easy

AUTHOR

Wolfdieter Lang, Jun 01 2007

STATUS

approved

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Last modified July 5 09:20 EDT 2015. Contains 259232 sequences.