A201950 Central coefficients in Product_{k=0..n-1} (1 + k*x + x^2).

1, 0, 2, 6, 28, 160, 1078, 8358, 73260, 716112, 7721844, 91039740, 1164932470, 16077368580, 238037983558, 3763371442530, 63276351409092, 1127406030014112, 21218146474666864, 420611921077524912, 8759617763834095796, 191208185756772875880

Offset

0,3

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Formula

Central terms of rows in irregular triangle A201949.

a(n) = (n-1)*a(n-1) + 2*A201952(n-2) for n>0.

E.g.f.: BesselI(0, 2*log(1 - x)). - Ilya Gutkovskiy, Feb 22 2019

E.g.f.: Sum_{n>=0} log(1 - x)^(2*n) / n!^2. [After Ilya Gutkovskiy - Paul D. Hanna, Feb 24 2019]

Example

The coefficients in Product_{k=0..n-1} (1+k*x+x^2) form triangle A201949:
(1);
1,(0), 1;
1, 1,(2), 1, 1;
1, 3, 5, (6), 5, 3, 1;
1, 6, 15, 24, (28), 24, 15, 6, 1;
1, 10, 40, 90, 139, (160), 139, 90, 40, 10, 1;
1, 15, 91, 300, 629, 945, (1078), 945, 629, 300, 91, 15, 1;
1, 21, 182, 861, 2520, 5019, 7377, (8358), 7377, 5019, 2520, 861, 182, 21, 1;
1, 28, 330, 2156, 8729, 23520, 45030, 65016, (73260), 65016, 45030, 23520, 8729, 2156, 330, 28, 1; ...
where coefficients in parenthesis form the initial terms of this sequence.

Mathematica

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

Prog

(PARI) {a(n) = polcoeff( prod(k=1, n, 1+(k-1)*x+x^2+x*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))
(PARI) /* From series BesselI(0, 2*log(1 - x)), after Ilya Gutkovskiy */
{a(n) = n!*polcoeff( sum(m=0, n, log(1 - x +x*O(x^n))^(2*m)/m!^2), n)}
for(n=0, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Feb 24 2019

Crossrefs

Cf. A201949, A201951, A201952, A201953.

Cf. A324304 (variant).

Sequence in context: A184695 A086633 A354311 * A358265 A356633 A109570

Adjacent sequences: A201947 A201948 A201949 * A201951 A201952 A201953

Keyword

nonn

Author

Paul D. Hanna, Dec 06 2011

Status

approved