A133308 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*7^i*8^(n-i), a(0)=1.

1, 8, 120, 2248, 47160, 1059976, 24958200, 607693640, 15175702200, 386555020552, 10004252294520, 262321706465736, 6953918939056440, 186059575955360136, 5018045415643478520, 136276936332343342152, 3723442515218861494200, 102281105054908404972040

Offset: 0

Author

Philippe Deléham, Oct 18 2007

Keywords

• nonn

Comments

Eighth column of array A103209.

The Hankel transform of this sequence is 56^C(n+1,2). - Philippe Deléham, Oct 28 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 0..675

Crossrefs

Cf. A000108, A060693, A103209, A103210, A103211.

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-30*x+1))/(14*x))) // G. C. Greubel, Feb 10 2018

Maple

a := n -> hypergeom([-n, n+1], [2], -7);
seq(round(evalf(a(n), 32)), n=0..15); # Peter Luschny, May 23 2014

Mathematica

CoefficientList[Series[(1-x-Sqrt[x^2-30*x+1])/(14*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

PARI

x='x+O('x^30); Vec((1-x-sqrt(x^2-30*x+1))/(14*x)) \\ G. C. Greubel, Feb 10 2018

Formula

G.f.: (1-z-sqrt(z^2-30*z+1))/(14*z).

a(n) = Sum_{k, 0<=k<=n} A088617(n,k)*7^k.

a(n) = Sum_{k, 0<=k<=n} A060693(n,k)*7^(n-k).

a(n) = Sum_{k, 0<=k<=n} C(n+k, 2k)7^k*C(k), C(n) given by A000108.

a(0)=1, a(n) = a(n-1) + 7*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007

Conjecture: (n+1)*a(n) + 15*(-2*n+1)*a(n-1) + (n-2)*a(n-2) = 0. - R. J. Mathar, May 23 2014

a(n) = hypergeom([-n, n+1], [2], -7). - Peter Luschny, May 23 2014

G.f.: 1/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - 7*x/(1 - 8*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017