A133309 - OEIS

%I #36 Jan 28 2025 20:52:06

%S 1,9,153,3249,77265,1968633,52546473,1450365921,41058670113,

%T 1185580310121,34783088255289,1033907690362257,31070005849929969,

%U 942384250116160857,28812102048874578249,887007207177728561601,27473495809057571051073,855518113376312857290441

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

%C Ninth column of array A103209.

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

%H Vincenzo Librandi, <a href="/A133309/b133309.txt">Table of n, a(n) for n = 0..200</a>

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

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

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

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

%F 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

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

%F Recurrence: (n+1)*a(n) = 17*(2*n-1)*a(n-1) - (n-2)*a(n-2). - _Vaclav Kotesovec_, Aug 13 2013

%F 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

%t Rest@ CoefficientList[ Series[(1-x-Sqrt[x^2-34*x+1])/16, {x, 0, 18}], x] (* _Robert G. Wilson v_, Oct 19 2007 *)

%t Table[-((3 I LegendreP[n, -1, 2, 17])/(2 Sqrt[2])), {n, 0, 20}] (* _Vaclav Kotesovec_, Aug 13 2013 *)

%o (PARI) my(x='x+O('x^30)); Vec((1-x-sqrt(x^2-34*x+1))/16) \\ _G. C. Greubel_, Feb 10 2018

%o (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

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

%K nonn

%O 0,2

%A _Philippe Deléham_, Oct 18 2007

%E More terms from _Robert G. Wilson v_, Oct 19 2007