A115202 Fifth column of triangle A115193 (called C(1,2)).

1, 9, 67, 477, 3363, 23741, 168451, 1202685, 8641539, 62470141, 454164483, 3319054333, 24371503107, 179736723453, 1330803769347, 9889323810813, 73733148770307, 551423090098173, 4135500638060547

Offset: 0

Author

Wolfdieter Lang, Feb 03 2006

Keywords

• nonn
• easy

Comments

Also one eighth of the fourth diagonal of triangle A115195, called Y(1,2).

Links

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

Programs

Mathematica

f[n_] := SeriesCoefficient[(1 - 9*x + 14*x^2 - (1 - 5*x + 2*x^2) Sqrt[1 - 8*x])/(16*x^4*(1 + x)), {x, 0, n}];
Table[f[n], {n, 0, 50}] (* G. C. Greubel, Feb 04 2016 *)

Formula

a(n)= A115195(3+n,1+n)/8, n>=0.

G.f.: (-1 + 3*x + (1- 5*x + 2*x^2)*c(2*x))/(4*(1+x)*x^3), with the o.g.f. c(x) of A000108 (Catalan).

a(n) = A115193(4+n,4), n>=0.

a(n) = (-1)^n*8^(n+2)*(binomial(1/2, n+3)*Hypergeometric2F1(1,n+5/2; n+4; -8) + 20*binomial(1/2, n+4)*Hypergeometric2F1(1,n+7/2; n+5; -8) + 32*binomial(1/2, n+5)*Hypergeometric2F1(1,n+9/2; n+6; -8)). - G. C. Greubel, Feb 04 2016

D-finite with recurrence (n+4)*a(n) +2*(-6*n-13)*a(n-1) +(29*n-10)*a(n-2) +2*(13*n+22)*a(n-3) +8*(-2*n+3)*a(n-4)=0. - R. J. Mathar, Mar 10 2022