A191511 E.g.f. 1 - cos(3*x)^(1/3). (even powers only)

3, 27, 1863, 259767, 63267723, 23850461907, 12872337567183, 9418588525038447, 8974900856105748243, 10799459611549296021387, 16014456358054037241378903, 28692834058049011948073522727, 61105982516981628849258186347163, 152570799245287136693700721604134467, 441413217492406160002632205611608461023

Offset

1,1

Links

Table of n, a(n) for n=1..15.

Formula

a(n)=2*(sum(m=2..2*n, ((sum(k=1..m-1, binomial(k,m-k-1)*(-1)^(k+1)*3^(2*n-2*m+k+1)*binomial(m+k-1,m-1)))*sum(j=1..m, ((sum(i=0..((j-1)/2), (j-2*i)^(2*n)*binomial(j,i)))*binomial(m,j)*(-1)^(n+m-j))/2^j))/(m)))-

((-1)^n*3^(2*n-1)), n>0.

a(n) ~ Gamma(1/3) * 2^(4*n - 2/3) * 3^(2*n - 1/2) * n^(2*n - 5/6) / (Pi^(2*n + 1/6) * exp(2*n)). - Vaclav Kotesovec, Jun 05 2019

Mathematica

nmax = 40; Table[(CoefficientList[Series[1 - Cos[3*x]^(1/3), {x, 0, nmax}], x] * Range[0, nmax]!)[[n]], {n, 3, nmax, 2}] (* Vaclav Kotesovec, Jun 05 2019 *)

Prog

(Maxima)
a(n):=2*(sum(((sum(binomial(k, m-k-1)*(-1)^(k+1)*3^(2*n-2*m+k+1)*binomial(m+k-1, m-1), k, 1, m-1))*sum(((sum((j-2*i)^(2*n)*binomial(j, i), i, 0, ((j-1)/2)))*binomial(m, j)*(-1)^(n+m-j))/2^j, j, 1, m))/(m), m, 2, 2*n))-
((-1)^n*3^(2*n-1));

Crossrefs

Sequence in context: A229866 A305843 A192341 * A102580 A051576 A184278

Adjacent sequences: A191508 A191509 A191510 * A191512 A191513 A191514

Keyword

nonn

Author

Vladimir Kruchinin, Jun 05 2011

Status

approved