A383704 a(n) = [x^n] Product_{k=0..2*n-1} (x - (-1)^k * (2*k+1)).

1, 2, -34, -540, 26614, 805980, -66399124, -2972817848, 343902030758, 20389669252524, -3039312653124540, -224361715353976200, 40941662601331486396, 3617518823154571788440, -781104190733806836937320, -80375840650247250199417200, 20044038897159722534821833990

Offset: 0

Author

Seiichi Manyama, May 06 2025

Keywords

• sign

Links

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

Crossrefs

Cf. A293318.

Programs

Magma

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

Mathematica

Table[Sum[(4*n-1)^k*4^(n-k)*Binomial[n+k,n]*StirlingS1[2*n,n+k],{k,0,n}],{n,0,17}] (* Vincenzo Librandi, May 07 2025 *)

PARI

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

Formula

a(n) = Sum_{k=0..n} (-(4*n-3))^k * 4^(n-k) * binomial(n+k,n) * |Stirling1(2*n,n+k)|.

a(n) = Sum_{k=0..n} (4*n-1)^k * 4^(n-k) * binomial(n+k,n) * Stirling1(2*n,n+k).

a(n) = (2*n)! * [x^(2*n)] 1/f(x)^(4*n-3) * log(f(x))^n / n!, where f(x) = 1/(1 - 4*x)^(1/4).