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A250917 Expansion of e.g.f. exp( x*C(x)^3 ) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, A000108.

%I A250917 #20 Mar 15 2025 11:29:15
%S A250917 1,1,7,73,1033,18541,403831,10351237,305355793,10192132153,
%T A250917 379819484551,15634219476481,704566985120857,34506514429777573,
%U A250917 1825081888365736183,103685565729559782781,6297505655719537293601,407233553972252986277617,27935786938445348562454663
%N A250917 Expansion of e.g.f. exp( x*C(x)^3 ) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, A000108.
%C A250917 In general, if k>0 and e.g.f. = exp(x*C(x)^k) where C(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. of the Catalan numbers, then a(n) ~ k * 2^(2*n + k - 5/2) * n^(n-1) / exp(n - 2^(k-2)). - _Vaclav Kotesovec_, Aug 22 2017
%F A250917 a(n) = Sum_{k=0..n} n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k)  for n>0 with a(0)=1.
%F A250917 a(n) ~ 3 * 2^(2*n+1/2) * n^(n-1) / exp(n-2). - _Vaclav Kotesovec_, Aug 22 2017
%F A250917 Conjecture D-finite with recurrence: +2*a(n) +(-11*n+20)*a(n-1) +(n^3+9*n^2-116*n+164)*a(n-2) +(-4*n^4+35*n^3+n^2-317*n+342)*a(n-3) -6*(n-3)*(6*n^3-50*n^2+147*n-176)*a(n-4) +12*(n-5)*(2*n-9)*(n-3)*(n-4)*a(n-5)=0. - _R. J. Mathar_, Jan 25 2020
%F A250917 E.g.f.: exp( (1/x)^2 * Series_Reversion( x*(1-x) )^3 ). - _Seiichi Manyama_, Mar 15 2025
%e A250917 E.g.f.: A(x) = 1 + x + 7*x^2/2! + 73*x^3/3! + 1033*x^4/4! + 18541*x^5/5! +...
%e A250917 such that log(A(x)) = x*C(x)^3,
%e A250917 log(A(x)) = x + 3*x^2 + 9*x^3 + 28*x^4 + 90*x^5 + 297*x^6 + 1001*x^7 +...
%e A250917 where C(x) = 1 + x*C(x)^2 is the g.f. of A000108.
%o A250917 (PARI) {a(n)=my(C=1); for(i=1, n, C=1+x*C^2 +x*O(x^n));
%o A250917 n!*polcoef(exp(x*C^3), n)}
%o A250917 for(n=0, 20, print1(a(n), ", "))
%o A250917 (PARI) {a(n) = if(n==0, 1, sum(k=0, n, n!/k! * binomial(2*n+k-1, n-k) * 3*k/(n+2*k) ))}
%o A250917 for(n=0, 20, print1(a(n), ", "))
%o A250917 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(serreverse(x*(1-x))^3/x^2))) \\ _Seiichi Manyama_, Mar 15 2025
%Y A250917 Cf. A080893, A251568.
%Y A250917 Cf. A091695, A380512, A380515.
%Y A250917 Cf. A000108, A001517.
%K A250917 nonn
%O A250917 0,3
%A A250917 _Paul D. Hanna_, Dec 06 2014