A386767 a(n) = Sum_{k=0..n} 5^k * 2^(n-k) * binomial(2*n+k-1,k).
1, 12, 294, 8178, 240186, 7271832, 224484684, 7024333608, 221997758346, 7069839153252, 226514542354974, 7293106513777338, 235771657829954856, 7648097463209959872, 248816951694728297664, 8115177647328907792368, 265257523746851227499466, 8687091365891501763853692
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 5^k*2^(n-k)*binomial(2*n+k-1, k));
Formula
a(n) = Sum_{k=0..n} 2^k * 3^(n-k) * binomial(3*n,k) * binomial(3*n-k-1,n-k).
a(n) = [x^n] ( (1+2*x)^3/(1-3*x)^2 )^n.
The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-3*x)^2 / (1+2*x)^3 ). See A386773.
a(n) = Sum_{k=0..n} 5^k * (-3)^(n-k) * binomial(3*n,k).
D-finite with recurrence 54*n*(2*n-1)*a(n) +3*(-2063*n^2+4087*n-2340)*a(n-1) +4*(21379*n^2-81239*n+73110)*a(n-2) +50*(277*n^2-6323*n+15702)*a(n-3) -8400*(3*n-10)*(3*n-11)*a(n-4)=0. - R. J. Mathar, Aug 03 2025