A129256 Central coefficient of Product_{k=0..n} (1+k*x)^2.

1, 2, 13, 144, 2273, 46710, 1184153, 35733376, 1251320145, 49893169050, 2232012515445, 110722046632560, 6032418472347265, 358103844593876654, 23007314730623658225, 1590611390957425536000, 117745011140615270168865

Offset

0,2

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..354

Formula

a(n) = (-1)^n*Sum_{k=0..n} Stirling1(n+1,k+1)*Stirling1(n+1,n-k+1). - Paul D. Hanna, Jul 16 2009

a(n) ~ c * d^n * (n-1)!, where d = A238261 = -(2*LambertW(-1,-exp(-1/2)/2))^2 / (1 + 2*LambertW(-1,-exp(-1/2)/2)) = 4.910814964568255..., c = (-LambertW(-1, -exp(-1/2)/2))^(3/2)/(sqrt(-1 - LambertW(-1, -exp(-1/2)/2))*Pi) = 0.851946112888790982829578047527831525434714038256... . - Vaclav Kotesovec, Feb 10 2015, updated May 14 2025

Example

This sequence equals the central terms of the triangle in which the g.f. of row n is (1+x)^2*(1+2x)^2*(1+3x)^2*...*(1+n*x)^2, as illustrated by:
  (1);
   1, (2),  1;
   1,  6, (13),  12,     4;
   1, 12,  58, (144),  193,    132,      36;
   1, 20, 170,  800, (2273),  3980,    4180,   2400,    576;
   1, 30, 395, 3000, 14523, (46710), 100805, 143700, 129076, 65760, 14400;
  ...

Mathematica

Flatten[{1, Table[Coefficient[Expand[Product[(1+k*x), {k, 0, n}]^2], x^n], {n, 1, 20}]}] (* Vaclav Kotesovec, Feb 10 2015 *)

Prog

(PARI) a(n)=polcoeff(prod(k=0, n, 1+k*x)^2, n)
(PARI) {a(n)=(-1)^n*sum(k=0, n, stirling(n+1, k+1, 1)*stirling(n+1, n-k+1, 1))} \\ Paul D. Hanna, Jul 16 2009

Crossrefs

Cf. A008275 (Stirling1 numbers), A187235, A238261, A246117, A254882, A350376.

Cf. A384012, A384031, A384017.

Sequence in context: A003326 A207493 A003581 * A046245 A178248 A389833

Adjacent sequences: A129253 A129254 A129255 * A129257 A129258 A129259

Keyword

nonn

Author

Paul D. Hanna, Apr 06 2007

Status

approved