OFFSET
0,3
COMMENTS
Also coefficient of x^n in the expansion of (1 + x + (n*x)^2)^n.
Also coefficient of x^n in the expansion of 1/sqrt(1 - 2*x + (1-4*n^2)*x^2).
LINKS
Seiichi Manyama, Table of n, a(n) for n = 0..351
FORMULA
a(n) = Sum_{k=0..n} (1-2*n)^(n-k) * n^k * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..n} (1+2*n)^(n-k) * (-n)^k * binomial(n,k) * binomial(2*k,k).
a(n) = Sum_{k=0..floor(n/2)} n^(2*k) *binomial(n,2*k) * binomial(2*k,k).
a(n) ~ (exp(1/2) + (-1)^n * exp(-1/2)) * 2^(n - 1/2) * n^(n - 1/2) / sqrt(Pi). - Vaclav Kotesovec, May 02 2019
MATHEMATICA
Flatten[{1, Table[Sum[(-1)^k * (2*n + 1)^(n-k) * n^k * Binomial[n, k] * Binomial[2*k, k], {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, May 02 2019 *)
PROG
(PARI) {a(n) = polcoef((n/x+1+n*x)^n, 0)}
(PARI) {a(n) = polcoef((1+x+(n*x)^2)^n, n)}
(PARI) {a(n) = sum(k=0, n, (1-2*n)^(n-k)*n^k*binomial(n, k)*binomial(2*k, k))}
(PARI) {a(n) = sum(k=0, n, (1+2*n)^(n-k)*(-n)^k*binomial(n, k)*binomial(2*k, k))}
(PARI) {a(n) = sum(k=0, n\2, n^(2*k)*binomial(n, 2*k)*binomial(2*k, k))}
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, May 01 2019
STATUS
approved