A303923 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^n - A(x) )^n.

1, 1, 1, 1, 2, 6, 22, 92, 419, 2066, 10863, 60459, 354381, 2177439, 13979759, 93527819, 650509643, 4694372980, 35086564926, 271174745565, 2164066408692, 17808271012127, 150925549288155, 1315804758238582, 11787981398487995, 108409978503340041, 1022519935940220983, 9882436548778410911, 97788364370359938816

Offset: 0

Author

Paul D. Hanna, May 03 2018

Keywords

• nonn

Comments

Compare to: 1 = Sum_{n>=0} ( 1 + x*G(x)^k - G(x) )^n holds trivially for fixed k>0 when G(x) = 1 + x*G(x)^k ; this sequence explores the case when k varies with n.

Examples

			G.f.: A(x) = 1 + x + x^2 + x^3 + 2*x^4 + 6*x^5 + 22*x^6 + 92*x^7 + 419*x^8 + 2066*x^9 + 10863*x^10 + 60459*x^11 + 354381*x^12 + ...
such that
1 = 1 + (1 + x*A(x) - A(x)) + (1 + x*A(x)^2 - A(x))^2 + (1 + x*A(x)^3 - A(x))^3 + (1 + x*A(x)^4 - A(x))^4 + (1 + x*A(x)^5 - A(x))^5 + ...
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A303924, A303925, A303926.

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)^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)^n - A(x) )^n.

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

G.f.: x/Series_Reversion( x*F(x) ) such that 1 = Sum_{n>=0} ((1 + x*F(x))^(n+1) - F(x))^n, where F(x) is the g.f. of A303924.

G.f.: sqrt( x/Series_Reversion( x*G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^(n+2) - G(x))^n, where G(x) is the g.f. of A303925.