A383222 Coefficient of x^4 in expansion of (x+2) * (x+5) * ... * (x+3*n-1).

0, 0, 0, 0, 1, 40, 1275, 39655, 1276009, 43382934, 1570298610, 60630265740, 2495678898636, 109326548645600, 5085420626585936, 250576924194171120, 13046999027750243984, 716156618057417103008, 41347880768363832470304, 2505655766070932929630464

Offset: 0

Author

Seiichi Manyama, Apr 20 2025

Keywords

• nonn
• easy

Crossrefs

Column k=4 of A225470.

Programs

PARI

a(n) = polcoef(prod(k=0, n-1, x+3*k+2), 4);

PARI

first(n) = {my(res = vector(n), v = [1, 0, 0, 0, 0], cv, c = -1, pc = 1); for(i = 2, n, c+=3; pc *= c; cv = v[^5]; cv = concat(0, cv); cv+=v*c; v = cv; res[i] = v[5]); res} \\ David A. Corneth, May 06 2025

Formula

a(n) = Sum_{k=4..n} 2^(k-4) * 3^(n-k) * binomial(k,4) * |Stirling1(n,k)|.

E.g.f.: f(x)^2 * log(f(x))^4 / 24, where f(x) = 1/(1 - 3*x)^(1/3).

a(n) = Sum_{k=4..n} (3*n-1)^(k-4) * 3^(n-k) * binomial(k,4) * Stirling1(n,k). - Seiichi Manyama, May 06 2025