A383702 Coefficient of x^2 in expansion of (x+3) * (x+7) * ... * (x+4*n-1).

0, 0, 1, 21, 446, 10670, 290599, 8951355, 308846124, 11822475564, 497794079421, 22881487815153, 1140642637297866, 61312161303209466, 3535773901817957955, 217787248332803277495, 14271822475100747003160, 991517953843097370650520, 72799719644532661375481145

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=2 of A225471.

Programs

Magma

[&+[(4*n-1)^(k-2) * 4^(n-k) * Binomial(k,2) * StirlingFirst(n,k): k in [0..n]]: n in [0..25]]; // Vincenzo Librandi, May 07 2025

Mathematica

Array[Residue[Product[(x + 4*k - 1), {k, 1, #}]/x^3, {x, 0}] &, 19, 0] (* Shenghui Yang, May 06 2025 *)
Table[Sum[(4*n-1)^(k-2)*4^(n-k)*Binomial[k,2]*StirlingS1[n,k],{k,0,n}],{n,0,18}] (* Vincenzo Librandi, May 07 2025 *)

PARI

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

Formula

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

a(n) = Sum_{k=2..n} (4*n-1)^(k-2) * 4^(n-k) * binomial(k,2) * Stirling1(n,k).

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

Conjecture D-finite with recurrence a(n) +3*(-4*n+5)*a(n-1) +(48*n^2-168*n+151)*a(n-2) -(4*n-9)^3*a(n-3)=0. - R. J. Mathar, May 07 2025