A383703 Coefficient of x^3 in expansion of (x+3) * (x+7) * ... * (x+4*n-1).

0, 0, 0, 1, 36, 1130, 36660, 1280419, 48644344, 2011398164, 90267003960, 4379275249701, 228707424551100, 12804721289403966, 765571832220427596, 48704512002823186119, 3286171504510664002992, 234445313277315235203624, 17637135196532479070107824, 1395584859384468591633567945

Offset: 0

Author

Seiichi Manyama, May 06 2025

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Column k=3 of A225471.

Programs

Magma

[&+[(4*n-1)^(k-3) * 4^(n-k) * Binomial(k,3) * StirlingFirst(n,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 07 2025

Mathematica

Table[Sum[(4*n-1)^(k-3)*4^(n-k)*Binomial[k,3]*StirlingS1[n,k],{k,3,n}],{n,0,25}] (* Vincenzo Librandi, May 07 2025 *)

PARI

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

Formula

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

a(n) = Sum_{k=3..n} (4*n-1)^(k-3) * 4^(n-k) * binomial(k,3) * Stirling1(n,k).

E.g.f.: f(x)^3 * log(f(x))^3 / 6, where f(x) = 1/(1 - 4*x)^(1/4).