%I #9 Mar 10 2022 05:54:15

%S 1,13,123,1037,8291,64509,494595,3761661,28486659,215277565,

%T 1625688067,12277764093,92783468547,701828038653,5314762113027,

%U 40297495658493,305941006516227,2325794003091453,17704219384479747

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

%C Also sixth diagonal of triangle A115195, called Y(1,2), divided by 32.

%H G. C. Greubel, <a href="/A115204/b115204.txt">Table of n, a(n) for n = 0..500</a>

%F a(n) = A115195(5+n,1+n)/32, n>=0.

%F G.f.: (-1 + 7*x - 8*x^2 + (1- 9*x + 18*x^2 - 4*x^3)*c(2*x))/(16*(1+x)*x^5), 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))^6)/(64*(1+x)*x^6).

%F a(n) = A115193(6+n,6), n>=0.

%F a(n) = (-1)^n*2^(8+3*n)*(Binomial[1/2, 4 + n]*Hypergeometric2F1[1, 7/2 + n, 5 + n, -8] + 4*(9*Binomial[1/2, 5 + n]*Hypergeometric2F1[1, 9/2 + n, 6 + n, -8] + 36*Binomial[1/2, 6 + n]*Hypergeometric2F1[1, 11/2 + n, 7 + n, -8] + 32*Binomial[1/2, 7 + n]*Hypergeometric2F1[1, 13/2 + n, 8 + n, -8])). - _G. C. Greubel_, Feb 04 2016

%F D-finite with recurrence 2*n*(n+6)*a(n) +(-11*n^2-51*n-120)*a(n-1) +(-37*n^2-99*n-132)*a(n-2) -12*(n+1)*(2*n+1)*a(n-3)=0. - _R. J. Mathar_, Mar 10 2022

%t f[n_] := SeriesCoefficient[(1 - 13*x + 46*x^2 - 36*x^3 -(1 - 9*x + 18*x^2 - 4*x^3) Sqrt[1 - 8*x])/(64*x^6*(1 + x)), {x, 0, n}];

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

%Y Cf. A000108, A115202, A115203.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_, Feb 03 2006