A341546 a(n) = Sum_{k=0..n} (-1)^(n+k)*A111516(n,k).

1, 0, 2, 12, 50, 280, 1442, 7812, 42338, 232176, 1280642, 7103932, 39579410, 221340808, 1241708834, 6984796852, 39382895810, 222512915680, 1259482604546, 7140546372204, 40541480041970, 230480474747640, 1311841695315362, 7474722997813732, 42631911134818850

Offset

0,3

Links

Table of n, a(n) for n=0..24.

Formula

O.g.f.: ((-3*x^2 + 5*x + 4)*sqrt(x^2 - 6*x + 1) - 3*x^3 + 2*x^2 + 5*x)/(sqrt(x^2 - 6*x + 1)*(6*x^2 + 10*x + 4)).

a(n) ~ sqrt(9*sqrt(2) - 8) * (1 + sqrt(2))^(2*n) / (14*sqrt(Pi*n)). - Vaclav Kotesovec, Feb 14 2021

D-finite with recurrence -2*(n-1)*(56*n^2-328*n+467)*a(n) +(392*n^3-3024*n^2+7501*n-5979)*a(n-1) +(1400*n^3-10328*n^2+24

187*n-17824)*a(n-2) +(728*n^3-5216*n^2+11919*n-8571)*a(n-3) -3*(n-4)*(56*n^2-216*n+195)*a(n-4)=0. - R. J. Mathar, Mar 06 2022

D-finite with recurrence 10*(-n+1)*a(n) +(31*n-53)*a(n-1) +(145*n-274)*a(n-2) +2*(47*n-69)*a(n-3) +2*(-32*n+179)*a(n-4) +3*(-15*n+77)*a(n-5) +9*(n-6)*a(n-6)=0. - R. J. Mathar, Mar 06 2022

Example

a(3) = (-1)^3*(1 - 7 + 12 - 18) = 12.
a(4) = (-1)^4*(1 - 15 + 32 - 56 + 88) = 50.

Prog

(PARI) G(n, k) = if (k==0, 1, sum(j=1, n, binomial(n, j)*binomial(k+j-2, j-1))); \\ A111516
a(n) = sum(k=0, n, (-1)^(n+k)*G(n, k)); \\ Michel Marcus, Feb 14 2021

Crossrefs

Cf. A111516.

Sequence in context: A115243 A218776 A241683 * A012423 A012427 A012695

Adjacent sequences: A341543 A341544 A341545 * A341547 A341548 A341549

Keyword

nonn

Author

Petros Hadjicostas, Feb 14 2021

Status

approved