A303927 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( (1 + x*A(x)^2)^n - A(x) )^n.

1, 1, 3, 19, 199, 2863, 51280, 1087107, 26492959, 728234294, 22273547313, 750180870861, 27591387247199, 1100527782602563, 47324815446060104, 2182852921566858499, 107515416285928793865, 5632697086212688424650, 312779421789041421062682, 18351511395587408908636348, 1134459736825581425674735933

Offset: 0

Author

Paul D. Hanna, May 03 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 19*x^3 + 199*x^4 + 2863*x^5 + 51280*x^6 + 1087107*x^7 + 26492959*x^8 + 728234294*x^9 + 22273547313*x^10 + ...
such that
1 = 1 + ((1 + x*A(x)^2) - A(x)) + ((1 + x*A(x)^2)^2 - A(x))^2 + ((1 + x*A(x)^2)^3 - A(x))^3 + ((1 + x*A(x)^2)^4 - A(x))^4 + ((1 + x*A(x)^2)^5 - A(x))^5 + ...
		

Crossrefs

Cf. A303926, A303928.

Programs

PARI

{a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ( (1 + x*Ser(A)^2)^m - Ser(A))^m ) )[#A] ); A[n+1]}
for(n=0,30, print1(a(n),", "))

Formula

G.f. A(x) satisfies:

(1) 1 = Sum_{n>=0} ( (1 + x*A(x)^2)^n - A(x) )^n.

(2) 1 = Sum_{n>=0} (1 + x*A(x)^2)^(n^2) / (1 + A(x)*(1 + x*A(x)^2)^n)^(n+1). - Paul D. Hanna, Dec 11 2018

G.f.: 1/x*Series_Reversion( x/F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^n - F(x))^n, where F(x) is the g.f. of A303926.

G.f.: x/Series_Reversion( x*G(x) ) such that 1 = Sum_{n>=0} ((1 + x*G(x)^3)^n - G(x))^n, where G(x) is the g.f. of A303928.