A386869 a(n) = Sum_{k=0..n} 2^(n-k) * binomial(3*n+2,k) * binomial(3*n-k,n-k).
1, 11, 168, 2839, 50333, 917604, 17036260, 320383295, 6082829067, 116342007859, 2238247173440, 43266114873636, 839661737871388, 16349646755219432, 319263686177979564, 6249714381417109903, 122603983720769666087, 2409746305286188995681
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 2^(n-k)*binomial(3*n+2, k)*binomial(3*n-k, n-k));
Formula
a(n) = [x^n] (1+x)^(3*n+2)/(1-2*x)^(2*n+1).
a(n) = [x^n] 1/((1-x)^2 * (1-3*x)^(2*n+1)).
a(n) = Sum_{k=0..n} 3^k * (-2)^(n-k) * (n-k+1) * binomial(3*n+2,k).
a(n) = Sum_{k=0..n} 3^k * (n-k+1) * binomial(2*n+k,k).
D-finite with recurrence 544*n*(2*n-1)*a(n) +8*(618*n^2-9184*n+8025)*a(n-1) +2*(-276538*n^2+1112059*n-1061145)*a(n-2) +15327*(3*n-4)*(3*n-5)*a(n-3)=0. - R. J. Mathar, Aug 19 2025