A133307 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*6^i*7^(n-i), a(0)=1.

1, 7, 91, 1477, 26845, 522739, 10663471, 224939113, 4866571801, 107393779423, 2407939176643, 54700070934061, 1256249370578293, 29119953189833611, 680401905145643863, 16008309928027493713, 378930780842531820721, 9017843351806985482423, 215634517504141993966891

Offset: 0

Author

Philippe Deléham, Oct 18 2007

Keywords

• nonn

Comments

Seventh column of array A103209.

The Hankel transform of this sequence is 42^C(n+1,2). - Philippe Deléham, Oct 28 2007

Links

• G. C. Greubel, Table of n, a(n) for n = 0..700

Crossrefs

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

Programs

Magma

Q:=Rationals(); R:=PowerSeriesRing(Q, 40); Coefficients(R!((1-x-Sqrt(x^2-26*x+1))/(12*x))) // G. C. Greubel, Feb 10 2018

Maple

a := n -> hypergeom([-n, n+1], [2], -6);
seq(round(evalf(a(n),32)),n=0..16); # Peter Luschny, May 23 2014

Mathematica

CoefficientList[Series[(1-x-Sqrt[x^2-26*x+1])/(12*x), {x,0,50}], x] (* G. C. Greubel, Feb 10 2018 *)

PARI

x='x+O('x^30); Vec((1-x-sqrt(x^2-26*x+1))/(12*x)) \\ G. C. Greubel, Feb 10 2018

Formula

G.f.: (1-z-sqrt(z^2-26*z+1))/(12*z).

a(n) = Sum_{k=0..n} A088617(n,k)*6^k .

a(n) = Sum_{k=0..n} A060693(n,k)*6^(n-k).

a(n) = Sum_{k=0..n} C(n+k, 2k)6^k*C(k), C(n) given by A000108.

a(0)=1, a(n) = a(n-1) + 6*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007

Conjecture: (n+1)*a(n) + 13*(-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], -6). # Peter Luschny, May 23 2014

G.f.: 1/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - 6*x/(1 - 7*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

a(n) ~ 42^(1/4) * (13 + 2*sqrt(42))^(n + 1/2) / (12*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Nov 29 2021