A115203 Sixth column of triangle A115193 (called C(1,2)).

1, 11, 93, 723, 5437, 40323, 297469, 2191875, 16164861, 119443459, 884719613, 6570430467, 48927031293, 365303660547, 2734459846653, 20518848036867, 154328140087293, 1163305103130627, 8787088644243453

Offset: 0

Author

Wolfdieter Lang, Feb 03 2006

Keywords

• nonn
• easy

Comments

Also fifth diagonal of triangle A115195, called Y(1,2), divided by 16.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..500

Crossrefs

Cf. A000108, A115202, A115204.

Programs

Mathematica

f[n_] := SeriesCoefficient[(1 - 11*x + 28*x^2 - 8*x^3 - (1 - 7*x + 8*x^2) Sqrt[1 - 8*x])/(32*x^5*(1 + x)), {x, 0, n}];
Table[f[n], {n, 0, 50}] (* G. C. Greubel, Feb 04 2016 *)

Formula

a(n) = A115195(4+n,1+n)/16, n>=0.

G.f.: (-1 + 5*x -2*x^2 + (1- 7*x + 8*x^2)*c(2*x))/(8*(1+x)*x^4) with the o.g.f. c(x) of A000108 (Catalan).

G.f. is also: ((1 + 2*x*c(2*x))*(2*x*c(2*x))^5)/(32*(1+x)*x^5).

a(n) = A115193(5+n,5), n>=0.

a(n) = (-1)^(n+1)* 2^(10 + 3*n)*(binomial(1/2,n+4)*Hypergeometric2F1(1, 7/2 + n, 5 + n, -8) + 7*binomial(1/2,n+5)*Hypergeometric2F1(1, 9/2 + n, 6 + n, -8) + 8*binomial(1/2,n+6)*Hypergeometric2F1(1, 11/2 + n, 7 + n, -8)). - G. C. Greubel, Feb 04 2016

a(n) ~ 2^(3*n+10) / (9*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Feb 05 2016

D-finite with recurrence (n+5)*a(n) +2*(-7*n-22)*a(n-1) +(49*n+43)*a(n-2) +124*a(n-3) +32*(-2*n+1)*a(n-4)=0. - R. J. Mathar, Mar 10 2022