OFFSET
0,2
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 0..400
FORMULA
a(n) = Sum_{k=0..floor(n/2)} n^(2*k) * binomial(2*(n-k),n-k) * binomial(n-k,k).
a(n) = Sum_{k=0..floor(n/2)} (n^2+1)^k * 2^(n-2*k) * binomial(n,2*k) * binomial(2*k,k).
a(n) = [x^n] (1 + 2*x + (n^2+1)*x^2)^n.
MATHEMATICA
Table[Sum[(n^2+1)^k*2^(n-2*k)*Binomial[n, 2*k]*Binomial[2*k, k], {k, 0, Floor[n/2]}], {n, 0, 30}] (* Vincenzo Librandi, Sep 01 2025 *)
PROG
(PARI) a(n) = sum(k=0, n\2, (n^2+1)^k*2^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));
(Magma) [(&+[n^(2*k) * Binomial(2*(n-k), n-k) * Binomial(n-k, k): k in [0..Floor(n/2)]]): n in [0..40]]; // Vincenzo Librandi, Sep 01 2025
CROSSREFS
KEYWORD
nonn
AUTHOR
Seiichi Manyama, Aug 29 2025
STATUS
approved