%I #9 Apr 05 2021 04:14:15
%S 1,1,2,11,80,714,7095,76206,864590,10227727,125001862,1568419058,
%T 20108619244,262510020319,3479914302802,46742907726147,
%U 635092339459857,8716058291255777,120686879727465365,1684357785848110976
%N Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^4 *x^k ] *x^n/n ).
%C Compare g.f. to a g.f. of the Catalan numbers (A000108):
%C . exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^2 *x^k ] *x^n/n ).
%H G. C. Greubel, <a href="/A181068/b181068.txt">Table of n, a(n) for n = 0..500</a>
%F a(n) ~ c * 16^n / n^(5/2), where c = 0.034183651246881715583041336040447549489320454248320978... - _Vaclav Kotesovec_, Apr 05 2021
%e G.f.: A(x) = 1 + x + 2*x^2 + 11*x^3 + 80*x^4 + 714*x^5 + 7095*x^6 +...
%e The logarithm begins:
%e log(A(x)) = x + 3*x^2/2 + 28*x^3/3 + 275*x^4/4 + 3126*x^5/5 + 37632*x^6/6 +...+ A181069(n)*x^n/n +...
%e which equals the series:
%e log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
%e + (1 + 2^4*x + 3^4*x^2 + 4^4*x^3 + 5^4*x^4 + 6^4*x^5 +...)*x^2/2
%e + (1 + 3^4*x + 6^4*x^2 + 10^4*x^3 + 15^4*x^4 + 21^4*x^5 +...)*x^3/3
%e + (1 + 4^4*x + 10^4*x^2 + 20^4*x^3 + 35^4*x^4 + 56^4*x^5 +...)*x^4/4
%e + (1 + 5^4*x + 15^4*x^2 + 35^4*x^3 + 70^4*x^4 + 126^4*x^5 +...)*x^5/5
%e + (1 + 6^4*x + 21^4*x^2 + 56^4*x^3 + 126^4*x^4 + 252^4*x^5 +...)*x^6/6
%e + (1 + 7^4*x + 28^4*x^2 + 84^4*x^3 + 210^4*x^4 + 462^4*x^5 +...)*x^7/7 + ...
%t With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1, k]^4*x^(n+k)/n, {k,0,m+2}], {n,1,m+1}]], {x,0,m}], x]] (* _G. C. Greubel_, Apr 05 2021 *)
%o (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^4*x^k)*x^m/m)+x*O(x^n)), n)}
%o (Magma)
%o m:=30;
%o R<x>:=PowerSeriesRing(Integers(), m);
%o Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^4*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // _G. C. Greubel_, Apr 05 2021
%o (Sage)
%o m=30;
%o def A181068_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( exp( sum( sum( binomial(n+k-1,k)^4*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
%o A181068_list(m) # _G. C. Greubel_, Apr 05 2021
%Y Cf. A000108, A181069 (log), A181066 (variant).
%K nonn
%O 0,3
%A _Paul D. Hanna_, Oct 08 2010