A386937 a(n) = Sum_{k=0..n} binomial(3*n+1,k) * binomial(2*n-k-1,n-k).
1, 5, 38, 325, 2934, 27314, 259356, 2496813, 24281510, 237978598, 2346750900, 23257207714, 231438363324, 2311082461380, 23146003391352, 232402586792061, 2338665721556742, 23579860411878110, 238157209512898500, 2409099858256570710, 24403155769842168660
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, binomial(3*n+1, k)*binomial(2*n-k-1, n-k));
Formula
a(n) = [x^n] (1+x)^(3*n+1)/(1-x)^n.
a(n) = [x^n] 1/((1-x)^(n+2) * (1-2*x)^n).
a(n) = Sum_{k=0..n} 2^k * (-1)^(n-k) * binomial(3*n+1,k) * binomial(2*n-k+1,n-k).
a(n) = Sum_{k=0..n} 2^k * binomial(n+k-1,k) * binomial(2*n-k+1,n-k).
D-finite with recurrence n*(n+1)*a(n) +42*n*(n-2)*a(n-1) +12*(-33*n^2+120*n-95)*a(n-2) +72*(-63*n^2+189*n-110)*a(n-3) +3456*(3*n-8)*(3*n-10)*a(n-4)=0. - R. J. Mathar, Aug 19 2025