A341109 - OEIS

A341109

a(n) = denominator(p(n, x)) / (n!*denominator(bernoulli(n, x))), where p(n, x) = Sum_{k=0..n} E2(n, k)*binomial(x + k, 2*n) / Product_{j=1..n} (j - x) and E2(n, k) are the second-order Eulerian numbers A201637.

1, 1, 2, 4, 8, 16, 96, 192, 1152, 768, 1536, 3072, 18432, 36864, 221184, 147456, 884736, 1769472, 10616832, 21233664, 637009920, 424673280, 2548039680, 5096079360, 152882380800, 61152952320, 366917713920, 81537269760, 163074539520, 326149079040, 1956894474240

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OFFSET

0,3

COMMENTS

The challenge is to characterize the sequence purely arithmetically, i.e., without reference to the Eulerian numbers or the Bernoulli polynomials.

LINKS

Table of n, a(n) for n=0..30.

András Bazsó and István Mező, On the coefficients of power sums of arithmetic progressions, J. Number Th., 153 (2015), 117-123.

J.-L. Chabert and P.-J. Cahen, Old problems and new questions around integer-valued polynomials and factorial sequences In: J. W. Brewer, S. Glaz, W. J. Heinzer, B. M. Olberding (eds), Multiplicative Ideal Theory in Commutative Algebra. Springer, Boston, MA., 2006.

Bernd C. Kellner and Jonathan Sondow, Power-Sum Denominators, arXiv:1705.03857 [math.NT] 2017, Amer. Math. Monthly.

FORMULA

a(n) = A053657(n+1)/(n!*A144845(n)).

a(n) = (n+1)*A163176(n+1)/A144845(n).

a(n) = A341108(n)/A318256(n).

a(n) = A341107(n)*A324369(n+1).

a(n) = A341108(n)/A324370(n+1).

a(n) = A341108(n)*A007947(n+1)/A144845(n).

a(n) = A341108(n)*A324369(n+1)/A195441(n).

prime(n) divides a(k) for k >= A036689(n).

2^(n-1) divides exactly a(n) for n >= 2.

MAPLE