A201826 Central coefficients in Product_{k=1..n} (1 + k*y + y^2).

1, 1, 4, 18, 100, 660, 5038, 43624, 422252, 4516380, 52885644, 672781824, 9238314358, 136175455234, 2144494356834, 35930786795040, 638168940129732, 11976278012219556, 236791150694618872, 4919643784275283480, 107152493449339765396, 2441410548192907949196

Offset

0,3

Links

Table of n, a(n) for n=0..21.

Formula

E.g.f.: 1/(1-x) * Sum_{n>=0} log(1 - x)^(2*n) / n!^2. - Paul D. Hanna, Mar 02 2019

Example

E.g.f.: A(x) = 1 + x + 4*x^2/2! + 18*x^3/3! + 100*x^4/4! + 660*x^5/5! + 5038*x^6/6! + 43624*x^7/7! + 422252*x^8/8! + 4516380*x^9/9! + 52885644*x^10/10! + ...
The coefficients in Product_{k=1..n} (1 + k*y + y^2), n>=0, form triangle A249790:
[1];
[1, 1, 1];
[1, 3, 4, 3, 1];
[1, 6, 14, 18, 14, 6, 1];
[1, 10, 39, 80, 100, 80, 39, 10, 1];
[1, 15, 90, 285, 539, 660, 539, 285, 90, 15, 1];
[1, 21, 181, 840, 2339, 4179, 5038, 4179, 2339, 840, 181, 21, 1];
[1, 28, 329, 2128, 8400, 21392, 36630, 43624, 36630, 21392, 8400, 2128, 329, 28, 1]; ...
in which the central terms of the rows form this sequence.

Mathematica

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

Prog

(PARI) {a(n) = polcoeff(prod(k=1, n, 1 + k*x + x^2 +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) {a(n) = n!*polcoeff( sum(m=0, n, log(1 - x +x*O(x^n))^(2*m)/m!^2 ) / (1 - x +x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Mar 02 2019

Crossrefs

Cf. A249790, A202474, A201950, A202476.

Sequence in context: A371543 A371723 A215522 * A327833 A350267 A064852

Adjacent sequences: A201823 A201824 A201825 * A201827 A201828 A201829

Keyword

nonn

Author

Paul D. Hanna, Dec 19 2011

Status

approved