A387467 a(n) = Sum_{k=0..n} (1-n*i)^k * (1+n*i)^(n-k) * binomial(n,k)^2, where i is the imaginary unit.
1, 2, 14, 128, 2566, 44752, 1523724, 39267328, 1893328966, 64541150912, 4029767542756, 170848520912896, 13100724115628956, 664175960969073152, 60396776494002647768, 3563049510869692907520, 374818464874078558810694, 25220474024437034383526912, 3012865557320147302034729844
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..400
Programs
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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
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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 *) -
PARI
a(n) = sum(k=0, n\2, (n^2+1)^k*2^(n-2*k)*binomial(n, 2*k)*binomial(2*k, k));
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.