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A184360 G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (n+1)!^2*(x/2)^n.

%I A184360 #10 Apr 08 2025 10:40:27
%S A184360 1,2,5,34,442,8638,229467,7862664,336468450,17579403622,1101881183359,
%T A184360 81669937516066,7070184169543820,707266516140720872,
%U A184360 80989516005804384644,10528134125581145088720,1542184766049169920609018
%N A184360 G.f.: A(x) = x/Series_Reversion(x*G(x)) where G(x) = Sum_{n>=0} (n+1)!^2*(x/2)^n.
%F A184360 G.f. satisfies: A(x) = G(x/A(x)) where A(x*G(x)) = G(x) = Sum_{n>=0} (n+1)!^2*(x/2)^n.
%F A184360 G.f. satisfies: [x^n] A(x)^(n+1)/(n+1) = (n+1)!^2/2^n = A184358(n).
%e A184360 G.f.: A(x) = 1 + 2*x + 5*x^2 + 34*x^3 + 442*x^4 + 8638*x^5 +...
%e A184360 A(x)^(1/2) = 1 + x + 2*x^2 + 15*x^3 + 204*x^4 + 4085*x^5 + 110128*x^6 +...+ A184361(n)*x^n +...
%e A184360 The g.f. of A184358 is G(x) = A(x*G(x)):
%e A184360 G(x) = 1 + 2*x + 9*x^2 + 72*x^3 + 900*x^4 + 16200*x^5 + 396900*x^6 +...+ (n+1)!^2*x^n/2^n +...
%o A184360 (PARI) {a(n)=polcoeff(x/serreverse(x*sum(m=0,n+1,(m+1)!^2*(x/2)^m)+x^2*O(x^n)),n)}
%Y A184360 Cf. A184361, A184358, A182958.
%K A184360 nonn
%O A184360 0,2
%A A184360 _Paul D. Hanna_, Jan 16 2011