A387401 a(n) = Sum_{k=0..n} (1-i)^k * (1+i)^(n-k) * binomial(n+1,k) * binomial(n+1,n-k), where i is the imaginary unit.
1, 4, 18, 80, 360, 1632, 7448, 34176, 157536, 728960, 3384128, 15754752, 73525504, 343870464, 1611288960, 7562801152, 35550504448, 167339022336, 788643765248, 3720901222400, 17573439614976, 83074892775424, 393056192851968, 1861155016212480, 8819174122700800, 41818448615636992
Offset: 0
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Programs
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Magma
[&+[2^(n-k) * Binomial(n+1,n-2*k) * Binomial(2*k+1,k): k in [0..Floor (n/2)]]: n in [0..35]]; // Vincenzo Librandi, Sep 04 2025
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Mathematica
Table[Sum[2^(n-k)*Binomial[n+1,n-2*k]*Binomial[2*k+1,k],{k,0,Floor[n/2]}],{n,0,30}] (* Vincenzo Librandi, Sep 04 2025 *) -
PARI
a(n) = sum(k=0, n\2, 2^(n-k)*binomial(n+1, n-2*k)*binomial(2*k+1, k));
Formula
n*(n+2)*a(n) = (n+1) * (2*(2*n+1)*a(n-1) + 4*n*a(n-2)) for n > 1.
a(n) = Sum_{k=0..floor(n/2)} 2^(n-k) * binomial(n+1,n-2*k) * binomial(2*k+1,k).
a(n) = [x^n] (1+2*x+2*x^2)^(n+1).
E.g.f.: exp(2*x) * BesselI(1, 2*sqrt(2)*x) / sqrt(2), with offset 1.
a(n) = (n+1) * A071356(n).