A386830 a(n) = Sum_{k=0..n} 3^k * 2^(n-k) * binomial(3*n+1,k) * binomial(3*n-k,n-k).
1, 18, 459, 12942, 382671, 11632248, 360048924, 11287595862, 357239123631, 11389281564978, 365227235524539, 11767662196724232, 380651590433357316, 12354006908520865008, 402088229127633026304, 13119017347331737771302, 428955765661154879370351
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 3^k*2^(n-k)*binomial(3*n+1, k)*binomial(3*n-k, n-k));
Formula
a(n) = [x^n] (1+3*x)^(3*n+1)/(1-2*x)^(2*n+1).
a(n) = [x^n] 1/((1-3*x) * (1-5*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 5^k * (-2)^(n-k) * binomial(3*n+1,k).
a(n) = Sum_{k=0..n} 5^k * 3^(n-k) * binomial(2*n+k,k).
Conjecture D-finite with recurrence +8*n*(2*n-3)*a(n) +6*(-108*n^2+207*n-80)*a(n-1) +405*(3*n-2)*(3*n-4)*a(n-2)=0. - R. J. Mathar, Aug 19 2025