cp's OEIS Frontend

A184359 Recurrence: Sum_{n>=0} a(n-k)*a(k) = (n+1)!^2/2^n.

%I A184359 #12 Apr 08 2025 10:42:48
%S A184359 1,1,4,32,410,7562,188736,6118296,249991926,12575954918,764125698224,
%T A184359 55189878377480,4674557178309388,458942541226822876,
%U A184359 51705551381013381112,6626012145599584408536,958371653002293850802814
%N A184359 Recurrence: Sum_{n>=0} a(n-k)*a(k) = (n+1)!^2/2^n.
%F A184359 Self-convolution equals A184358.
%F A184359 G.f. satisfies: A(x) = F(x*A(x)^2) where A(x/F(x)^2) = F(x) is the g.f. of A184361.
%F A184359 G.f.: A(x) = sqrt((1/x)*Series_Reversion(x/F(x)^2)) where F(x) is the g.f. of A184361.
%e A184359 G.f.: A(x) = 1 + x + 4*x^2 + 32*x^3 + 410*x^4 + 7562*x^5 +...
%e A184359 A(x)^2 = 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 +...
%e A184359 The g.f. of A184361 is F(x) = A(x/F(x)^2):
%e A184359 F(x) = 1 + x + 2*x^2 + 15*x^3 + 204*x^4 + 4085*x^5 + 110128*x^6 +...
%o A184359 (PARI) {a(n)=local(G=sum(m=0,n,(m+1)!^2*x^m/2^m)+x*O(x^n));polcoeff(sqrt(G),n)}
%Y A184359 Cf. A184358, A184360, A184361.
%K A184359 nonn
%O A184359 0,3
%A A184359 _Paul D. Hanna_, Jan 16 2011