A115203 - OEIS

%I #11 Mar 10 2022 05:56:28

%S 1,11,93,723,5437,40323,297469,2191875,16164861,119443459,884719613,

%T 6570430467,48927031293,365303660547,2734459846653,20518848036867,

%U 154328140087293,1163305103130627,8787088644243453

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

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

%H G. C. Greubel, <a href="/A115203/b115203.txt">Table of n, a(n) for n = 0..500</a>

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

%F 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).

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

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

%F 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

%F a(n) ~ 2^(3*n+10) / (9*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Feb 05 2016

%F 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

%t 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}];

%t Table[f[n], {n, 0, 50}] (* _G. C. Greubel_, Feb 04 2016 *)

%Y Cf. A000108, A115202, A115204.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Feb 03 2006