A078794 a(n) = (-1)^(n+1) * Sum_{k=0..n} 16^k * B(2k) * C(2n+1,2k) where B(k) is the k-th Bernoulli number.

9, 15, 441, 12447, 555753, 35135919, 2990414745, 329655706431, 45692713833417, 7777794952987983, 1595024111042171769, 387863354088927172575, 110350957750914345093801, 36315529600705266098580207

Offset

1,1

Comments

For any m>0, sum(k=0,n,4^(m*k)*B(2*k)*C(2*n+1,2*k)) is always an integer. sum(k=0,n,4^k*B(2*k)*C(2*n+1,2*k)) = 2n+1.

Links

Table of n, a(n) for n=1..14.

Formula

It seems that a(n) is asymptotic to (n!)^2*w*z^n where z = 1.63....and w = ? [There is a missing factor sqrt(n), z = 16/Pi^2 = 1.6211389382774... - Vaclav Kotesovec, Feb 15 2019]

a(n) ~ (n!)^2 * 2^(4*n + 3) * sqrt(n) / Pi^(2*n + 3/2). - Vaclav Kotesovec, Feb 15 2019

Mathematica

Table[(-1)^(n+1)*Sum[16^k*BernoulliB[2*k]*Binomial[2*n + 1, 2*k], {k, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Feb 15 2019 *)

Prog

(PARI) a(n)=(-1)^(n+1)*sum(k=0, n, bernfrac(2*k)*binomial(2*n+1, 2*k)*16^k)

Crossrefs

Sequence in context: A158789 A355654 A100241 * A093595 A330429 A215418

Adjacent sequences: A078791 A078792 A078793 * A078795 A078796 A078797

Keyword

nonn

Author

Benoit Cloitre, Jan 10 2003

Status

approved