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

0,2

Links

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

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).

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 *)

Prog

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

Crossrefs

Cf. A293318.

Sequence in context: A104898 A218432 A071799 * A273052 A098704 A119298

Adjacent sequences: A383701 A383702 A383703 * A383705 A383706 A383707

Keyword

sign

Author

Seiichi Manyama, May 06 2025

Status

approved