OFFSET
0,4
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..635
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
FORMULA
a(n) = Sum_{k=0..floor(n/3)} (n-2*k)^(2*k) * binomial(n-2*k,k).
a(n) ~ exp(exp(4/3)*n^(1/3)/3^(1/3)) * n^(2*n/3) / 3^(2*n/3 + 1) * (1 + (3^(1/3)/(8*exp(4/3)) - 13*exp(8/3)/(6*3^(2/3))) / n^(1/3) + (67/(128*3^(1/3)*exp(8/3)) - 5*3^(2/3)*exp(4/3)/16 + 169*exp(16/3)/(216*3^(1/3))) / n^(2/3) + (3929/2304 + 497/(1024*exp(4)) + 7913*exp(4)/1728 - 2197*exp(8)/11664)/n). - Vaclav Kotesovec, Feb 19 2023
MATHEMATICA
Join[{1}, Table[Sum[Binomial[n - 2*k, k] * (n - 2*k)^(2*k), {k, 0, n/3}], {n, 1, 30}]] (* Vaclav Kotesovec, Feb 19 2023 *)
PROG
(PARI) my(N=30, x='x+O('x^N)); Vec(sum(k=0, N, (x*(1+(k*x)^2))^k))
(PARI) a(n) = sum(k=0, n\3, (n-2*k)^(2*k)*binomial(n-2*k, k));
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Feb 19 2023
STATUS
approved