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A216317 G.f.: 1/( (1-8*x)*(1+x)^2 )^(1/6). 2

%I #9 Jul 18 2013 03:48:08

%S 1,1,6,34,217,1449,9996,70512,505674,3672682,26943748,199284540,

%T 1483955746,11113108930,83628685440,631963708200,4793067536265,

%U 36469419494985,278278625232510,2128794954411930,16322188021280505,125405739232800585,965313101906567700

%N G.f.: 1/( (1-8*x)*(1+x)^2 )^(1/6).

%H Vincenzo Librandi, <a href="/A216317/b216317.txt">Table of n, a(n) for n = 0..200</a>

%F G.f.: exp( Sum_{n>=1} A007613(n)/2 * x^n/n ) where A007613(n) = Sum_{k=0..n} binomial(3*n,3*k).

%F Recurrence: n*a(n) = (7*n-6)*a(n-1) + 4*(2*n-3)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012

%F a(n) ~ Gamma(5/6)*8^n/(3^(2/3)*Pi*n^(5/6)). - _Vaclav Kotesovec_, Oct 20 2012

%e G.f.: A(x) = 1 + x + 6*x^2 + 34*x^3 + 217*x^4 + 1449*x^5 + 9996*x^6 +...

%e where 1/A(x)^6 = 1 - 6*x - 15*x^2 - 8*x^3.

%e The logarithm of the g.f. begins:

%e log(A(x)) = x + 11*x^2/2 + 85*x^3/3 + 683*x^4/4 + 5461*x^5/5 + 43691*x^6/6 +...+ A007613(n)/2*x^n/n +...

%t CoefficientList[Series[1/((1-8*x)*(1+x)^2)^(1/6), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Oct 20 2012 *)

%o (PARI) {a(n)=polcoeff(1/( (1-8*x)*(1+x)^2 +x*O(x^n) )^(1/6),n)}

%o (PARI) {a(n)=local(A=1+x); A=exp(sum(m=1, n+1, sum(j=0, m, binomial(3*m, 3*j))/2*x^m/m +x*O(x^n))); polcoeff(A, n)}

%o for(n=0, 31, print1(a(n), ", "))

%Y Cf. A216316, A007613.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Sep 03 2012

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Last modified February 22 22:43 EST 2024. Contains 370265 sequences. (Running on oeis4.)