A386866 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+2,k) * binomial(3*n-k-1,n-k).
1, 9, 132, 2197, 38649, 701292, 12979360, 243541725, 4616122851, 88173726337, 1694554311888, 32728267058604, 634701136059532, 12351249029265816, 241061116082196072, 4716751239386395885, 92494719333403946583, 1817328001770278062299, 35768122814759119268788
Offset: 0
Programs
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Mathematica
Table[Sum[2^(n-k) Binomial[3n+2,k]Binomial[3n-k-1,n-k],{k,0,n}],{n,0,20}] (* Harvey P. Dale, Sep 02 2025 *) -
PARI
a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+2, k)*binomial(3*n-k-1, n-k));
Formula
a(n) = [x^n] (1+x)^(3*n+2)/(1-2*x)^(2*n).
a(n) = [x^n] 1/((1-x)^3 * (1-3*x)^(2*n)).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * binomial(3*n+2,k) * binomial(n-k+2,n-k).
a(n) = Sum_{k=0..n} 3^k * binomial(2*n+k-1,k) * binomial(n-k+2,n-k).