A303924 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+1) - A(x) )^n.

1, 1, 2, 5, 15, 52, 204, 891, 4266, 22092, 122358, 718282, 4438154, 28711805, 193700970, 1358588449, 9883071724, 74423630202, 579231718432, 4652864427983, 38528749877802, 328519744186940, 2881366257269722, 25969840412367362, 240307819488203558, 2280902112035109237, 22187847195528993904, 221024332987155498348

Offset: 0

Author

Paul D. Hanna, May 03 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 2*x^2 + 5*x^3 + 15*x^4 + 52*x^5 + 204*x^6 + 891*x^7 + 4266*x^8 + 22092*x^9 + 122358*x^10 + 718282*x^11 + 4438154*x^12 + ...
such that
1 = 1 + (1 + x*A(x)^2 - A(x)) + (1 + x*A(x)^3 - A(x))^2 + (1 + x*A(x)^4 - A(x))^3 + (1 + x*A(x)^5 - A(x))^4 + (1 + x*A(x)^6 - A(x))^5 + ...
		

Links

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

Crossrefs

Cf. A303923, A303925.

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+1) - 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+1) - A(x) )^n.

(2) 1 = Sum_{n>=0} x^n * A(x)^(n*(n+1)) / (1 + (A(x)-1)*A(x)^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 A303923.

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