A061053 a(n) = (n+1)!*Sum_{k=0..n} C(2k, k)*B(n-k), where B(n) is n-th Bernoulli number.

1, 3, 31, 416, 7316, 158592, 4079832, 121418880, 4102640064, 155127605760, 6488944560000, 297483185986560, 14831664692912640, 798949604318423040, 46240823333993702400, 2861614126455843225600, 188557593322666066329600

Offset

0,2

Comments

The 1st negative term is a(64) = -1461516... (130 digits).

It appears that for n >= 64, a(n) < 0 if and only if n == 0 or 1 (mod 4). - Robert Israel, Sep 21 2015

Links

Harry J. Smith and Robert Israel, Table of n, a(n) for n = 0..280 (n = 0..100 from Harry J. Smith)

Example

a(3) = 4! *(binomial(0,0) B_3 + binomial(2,1) B_2 + binomial(4,2) B_1 + binomial(6,3) B_0) = 24 *(1 *0 + 2 *(1/6) + 6 *(-1/2) + 20 *1) = 416.

Maple

f:= n -> (n+1)!*add(binomial(2*k, k)*bernoulli(n-k), k=0..n):
map(f, [$0..100]); # Robert Israel, Sep 21 2015

Mathematica

Table[(n + 1)! Sum[Binomial[2 k, k] BernoulliB[n - k], {k, 0, n}], {n,
0, 16}] (* Michael De Vlieger, Sep 21 2015 *)

Prog

(PARI) { default(realprecision, 500); for (n=0, 100, a=(n + 1)!*sum(k=0, n, binomial(2*k, k)*bernreal(n - k)); write("b061053.txt", n, " ", round(a)) ) } \\ Harry J. Smith, Jul 17 2009

Crossrefs

Sequence in context: A363529 A123818 A087591 * A047798 A349038 A126346

Adjacent sequences: A061050 A061051 A061052 * A061054 A061055 A061056

Keyword

easy,sign

Author

Leroy Quet, May 26 2001

Status

approved