A133309 a(n) = (1/n)*Sum_{i=0..n-1} C(n,i)*C(n,i+1)*8^i*9^(n-i), a(0)=1.

1, 9, 153, 3249, 77265, 1968633, 52546473, 1450365921, 41058670113, 1185580310121, 34783088255289, 1033907690362257, 31070005849929969, 942384250116160857, 28812102048874578249, 887007207177728561601, 27473495809057571051073, 855518113376312857290441

Offset

0,2

Comments

Ninth column of array A103209.

The Hankel transform of this sequence is 72^C(n+1,2). - Philippe Deléham, Oct 29 2007

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Formula

G.f.: (1-z-sqrt(z^2-34*z+1))/16.

a(n) = Sum_{k=0..n} A088617(n,k)*8^k.

a(n) = Sum_{k=0..n} A060693(n,k)*8^(n-k).

a(n) = Sum_{k=0..n} C(n+k, 2k)8^k*C(k), C(n) given by A000108.

a(0)=1, a(n) = a(n-1) + 8*Sum_{k=0..n-1} a(k)*a(n-1-k). - Philippe Deléham, Oct 23 2007

a(n) ~ sqrt(144+102*sqrt(2))*(17+12*sqrt(2))^n/(16*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 13 2013

Recurrence: (n+1)*a(n) = 17*(2*n-1)*a(n-1) - (n-2)*a(n-2). - Vaclav Kotesovec, Aug 13 2013

G.f.: 1/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - 8*x/(1 - 9*x/(1 - ...)))))), a continued fraction. - Ilya Gutkovskiy, May 10 2017

Mathematica

Rest@ CoefficientList[ Series[(1-x-Sqrt[x^2-34*x+1])/16, {x, 0, 18}], x] (* Robert G. Wilson v, Oct 19 2007 *)
Table[-((3 I LegendreP[n, -1, 2, 17])/(2 Sqrt[2])), {n, 0, 20}] (* Vaclav Kotesovec, Aug 13 2013 *)

Prog

(PARI) x='x+O('x^30); Vec((1-x-sqrt(x^2-34*x+1))/16) \\ G. C. Greubel, Feb 10 2018
(Magma) Q:=Rationals(); R<x>:=PowerSeriesRing(Q, 40); Coefficients(R!(((1 -x-Sqrt[x^2-34*x+1])/16)) // G. C. Greubel, Feb 10 2018

Crossrefs

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

Sequence in context: A169958 A012017 A130980 * A228713 A151835 A217822

Adjacent sequences: A133306 A133307 A133308 * A133310 A133311 A133312

Keyword

nonn

Author

Philippe Deléham, Oct 18 2007

Extensions

More terms from Robert G. Wilson v, Oct 19 2007

Status

approved