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A281817 a(n) = 2*Sum_{k odd} k!*Stirling2(n,k)/(k + 1). 0
0, 1, 1, 4, 19, 116, 871, 7764, 80179, 941812, 12403711, 181056404, 2901669739, 50656307508, 956922611191, 19449063226324, 423206168046499, 9816562636678004, 241805428075379311, 6303793707327637524, 173401707643671303259 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Recall the result Sum_{k = 0..n} (-1)^k*k!*Stirling2(n,k)/(k + 1) = Bernoulli(n) = A027641(n)/A027642(n). We can write this result as Bernoulli(n) = S_1(n) - S_2(n), where S1 = Sum_{k even} k!*Stirling2(n,k)/(k + 1) and S2 = Sum_{k odd} k!* Stirling2(n,k)/(k + 1). Here we record the values of the sums 2*S_2(n), which are easily seen to be integers.

The numbers a(n) are derived from a formula for the numbers Bernoulli(n). Surprisingly, there also appears to be a connection between a(2*n) and Bernoulli(2*n - 2): we conjecture a(2*n) - 1 = integer * the denominator of Bernoulli(2*n - 2) = integer * (Product_{p prime, p - 1 | 2*n - 2} p) (checked up to n = 200). For example, a(14) - 1 = 956922611190 is divisible by 2*3*5*7*13 where 2, 3, 5, 7 and 13 are the primes p such that p - 1 divides 12, while a(18) - 1 = 241805428075379310 is divisible by 2*3*5*17 where 2, 3, 5 and 17 are the primes p such that p - 1 divides 16.

The same result also appears to hold for the integer sequence b(n) := 2*Sum_{k odd} (-1)^((k-1)/2)*k!*Stirling2(n,k)/(k + 1).

LINKS

Table of n, a(n) for n=0..20.

Bai-Ni Guo, István Mező, Feng Qi, An explicit formula for Bernoulli polynomials in terms of r-Stirling numbers of the second kind, arxiv:1402.2340v1 [math.CO], 2014.

FORMULA

E.g.f.: ( -x - log(2 - exp(x)) )/(exp(x) - 1) = x + x^2/2! + 4*x^3/3! + 19*x^4/4! + 116*x^5/5! + .... (use the first equation on page 3 of Guo et al. with r = 0 and s = 1).

For prime p, a(p) = 1 (mod p). Conjecture: for prime p, a(2*p) = 1 (mod p).

MAPLE

seq(add((2*k+1)!*Stirling2(n, 2*k+1)/(k + 1), k = 0..floor((n-1)/2)), n = 0..20);

CROSSREFS

Cf. A000629, A008277, A027641, A027642, A027760.

Sequence in context: A209673 A261497 A217989 * A060907 A245505 A247056

Adjacent sequences:  A281814 A281815 A281816 * A281818 A281819 A281820

KEYWORD

nonn,easy

AUTHOR

Peter Bala, Jan 31 2017

STATUS

approved

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Last modified January 18 11:33 EST 2019. Contains 319271 sequences. (Running on oeis4.)