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%I #18 Jul 02 2023 09:35:13
%S 1,2,13,80,538,3740,26650,193160,1417945,10511450,78533629,590485208,
%T 4463274232,33886781840,258260802232,1974759985952,15143163422794,
%U 116417053435316,896996316176290,6925241271855296,53562550587963052,414948608904171464,3219356873886333676
%N G.f.: 1/( (1-8*x)*(1+x)^2 )^(1/3).
%H Vincenzo Librandi, <a href="/A216316/b216316.txt">Table of n, a(n) for n = 0..1000</a>
%F G.f.: exp( Sum_{n>=1} A007613(n)*x^n/n ), where A007613(n) = Sum_{k=0..n} binomial(3*n,3*k).
%F Recurrence: n*a(n) = (7*n-5)*a(n-1) + 8*(n-1)*a(n-2). - _Vaclav Kotesovec_, Oct 20 2012
%F a(n) ~ Gamma(2/3)*2^(3*n+1)/(3^(5/6)*Pi*n^(2/3)). - _Vaclav Kotesovec_, Oct 20 2012
%F Inverse binomial transform of A004987. - _Peter Bala_, Jul 02 2023
%e G.f.: A(x) = 1 + 2*x + 13*x^2 + 80*x^3 + 538*x^4 + 3740*x^5 + 26650*x^6 +...
%e where 1/A(x)^3 = 1 - 6*x - 15*x^2 - 8*x^3.
%e The logarithm of the g.f. begins:
%e log(A(x)) = 2*x + 22*x^2/2 + 170*x^3/3 + 1366*x^4/4 + 10922*x^5/5 + 87382*x^6/6 +...+ A007613(n)*x^n/n +...
%t CoefficientList[Series[1/((1-8*x)*(1+x)^2)^(1/3), {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) )^(1/3),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))*x^m/m +x*O(x^n))); polcoeff(A, n)}
%o for(n=0, 31, print1(a(n), ", "))
%Y Cf. A004987, A007613, A216317, A216357, A216358.
%K nonn,easy
%O 0,2
%A _Paul D. Hanna_, Sep 03 2012
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