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A181066 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^3 *x^k ] *x^n/n ).

%I A181066 #9 Apr 05 2021 03:53:38
%S A181066 1,1,2,7,31,157,865,5051,30774,193669,1250319,8240232,55239187,
%T A181066 375624781,2585449450,17982937876,126222946496,893073250063,
%U A181066 6363674671524,45631735776036,329065051395940,2385126419825231
%N A181066 Expansion of g.f.: exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^3 *x^k ] *x^n/n ).
%C A181066 Compare g.f. to a g.f. of the Catalan numbers (A000108):
%C A181066 . exp( Sum_{n>=1} [ Sum_{k>=0} C(n+k-1,k)^2 *x^k ] *x^n/n ).
%H A181066 G. C. Greubel, <a href="/A181066/b181066.txt">Table of n, a(n) for n = 0..500</a>
%e A181066 G.f.: A(x) = 1 + x + 2*x^2 + 7*x^3 + 31*x^4 + 157*x^5 + 865*x^6 +...
%e A181066 The logarithm begins:
%e A181066 log(A(x)) = x + 3*x^2/2 + 16*x^3/3 + 95*x^4/4 + 606*x^5/5 + 4032*x^6/6 +...+ A181067(n)*x^n/n +...
%e A181066 which equals the series:
%e A181066 log(A(x)) = (1 + x + x^2 + x^3 + x^4 + x^5 + x^6 +...)*x
%e A181066 + (1 + 2^3*x +  3^3*x^2 +  4^3*x^3 +   5^3*x^4 +   6^3*x^5 + ...)*x^2/2
%e A181066 + (1 + 3^3*x +  6^3*x^2 + 10^3*x^3 +  15^3*x^4 +  21^3*x^5 + ...)*x^3/3
%e A181066 + (1 + 4^3*x + 10^3*x^2 + 20^3*x^3 +  35^3*x^4 +  56^3*x^5 + ...)*x^4/4
%e A181066 + (1 + 5^3*x + 15^3*x^2 + 35^3*x^3 +  70^3*x^4 + 126^3*x^5 + ...)*x^5/5
%e A181066 + (1 + 6^3*x + 21^3*x^2 + 56^3*x^3 + 126^3*x^4 + 252^3*x^5 + ...)*x^6/6
%e A181066 + (1 + 7^3*x + 28^3*x^2 + 84^3*x^3 + 210^3*x^4 + 462^3*x^5 + ...)*x^7/7 + ...
%t A181066 With[{m=30}, CoefficientList[Series[Exp[Sum[Sum[Binomial[n+k-1, k]^3*x^k*x^n/n, {k, 0, m+2}], {n, m+1}]], {x,0,m}], x]] (* _G. C. Greubel_, Apr 05 2021 *)
%o A181066 (PARI) {a(n)=polcoeff(exp(sum(m=1, n, sum(k=0, n, binomial(m+k-1,k)^3*x^k)*x^m/m)+x*O(x^n)), n)}
%o A181066 (Magma)
%o A181066 m:=30;
%o A181066 R<x>:=PowerSeriesRing(Integers(), m);
%o A181066 Coefficients(R!( Exp( (&+[ (&+[ Binomial(n+k-1,k)^3*x^(n+k)/n : k in [0..m+2]]): n in [1..m+1]]) ) )); // _G. C. Greubel_, Apr 05 2021
%o A181066 (Sage)
%o A181066 m=30;
%o A181066 def A181066_list(prec):
%o A181066     P.<x> = PowerSeriesRing(ZZ, prec)
%o A181066     return P( exp( sum( sum( binomial(n+k-1,k)^3*x^(n+k)/n for k in (0..m+2) ) for n in (1..m+1)) ) ).list()
%o A181066 A181066_list(m) # _G. C. Greubel_, Apr 05 2021
%Y A181066 Cf. A000108, A181067 (log), A181068 (variant).
%K A181066 nonn
%O A181066 0,3
%A A181066 _Paul D. Hanna_, Oct 03 2010