A386900 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n+1,k).
1, 14, 235, 4178, 76495, 1426184, 26922076, 512838410, 9837067951, 189729498350, 3675700225435, 71474375851640, 1394164222173700, 27266825345422352, 534510606516137920, 10499123975453808698, 206595710100771337327, 4071693103719194746250
Offset: 0
Keywords
Programs
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Mathematica
Table[Sum[3^k*2^(n-k)*Binomial[3*n+1, k], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Aug 07 2025 *) -
PARI
a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(3*n+1, k));
Formula
a(n) = [x^n] (1+3*x)^(3*n+1)/(1-2*x).
a(n) = [x^n] 1/((1-3*x)^(2*n+1) * (1-5*x)).
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(3*n-k,n-k).
a(n) ~ 3^(4*n + 5/2) / (sqrt(Pi*n) * 2^(2*n+3)). - Vaclav Kotesovec, Aug 07 2025