A372370 Coefficient of x^n in the expansion of ( (1+x+x^2)^2 / (1+x) )^n.
1, 1, 5, 13, 53, 176, 677, 2451, 9333, 34978, 133580, 508806, 1953701, 7509178, 28981643, 112046213, 434289525, 1686080622, 6557830310, 25542229740, 99622788428, 389023326600, 1520817551742, 5951305115982, 23310374278437, 91380414955176, 358506409488102
Offset: 0
Keywords
Programs
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Mathematica
a[n_]:=SeriesCoefficient[((1+x+x^2)^2/(1+x))^n,{x,0,n}]; Array[a,27,0] (* Stefano Spezia, Apr 30 2024 *) -
PARI
a(n, s=2, t=2, u=-1) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));
Formula
a(n) = Sum_{k=0..floor(n/2)} binomial(2*n,k) * binomial(n-k,n-2*k).
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1+x) / (1+x+x^2)^2 ).