A303925 G.f. A(x) satisfies: 1 = Sum_{n>=0} ( 1 + x*A(x)^(n+2) - A(x) )^n.

1, 1, 3, 12, 56, 288, 1587, 9222, 55957, 352267, 2290842, 15343839, 105634437, 746478622, 5409932286, 40189454704, 305972524737, 2387238374532, 19090018863000, 156496468777604, 1315509548959765, 11341506519584442, 100300906407392783, 909967403153604712, 8468614126450656268, 80832677102193209308, 791071858022525348235

Offset: 0

Author

Paul D. Hanna, May 03 2018

Keywords

• nonn

Examples

			G.f.: A(x) = 1 + x + 3*x^2 + 12*x^3 + 56*x^4 + 288*x^5 + 1587*x^6 + 9222*x^7 + 55957*x^8 + 352267*x^9 + 2290842*x^10 + 15343839*x^11 + ...
such that
1 = 1 + (1 + x*A(x)^3 - A(x)) + (1 + x*A(x)^4 - A(x))^2 + (1 + x*A(x)^5 - A(x))^3 + (1 + x*A(x)^6 - A(x))^4 + (1 + x*A(x)^7 - A(x))^5 + ...
		

Links

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

Crossrefs

Cf. A303923, A303924.

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+2) - 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+2) - A(x) )^n.

(2) 1 = Sum_{n>=0} x^n * A(x)^(n*(n+2)) / (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+1) - F(x))^n, where F(x) is the g.f. of A303924.

G.f.: sqrt( 1/x*Series_Reversion( x/G(x)^2 ) ) such that 1 = Sum_{n>=0} ((1 + x*G(x))^n - G(x))^n, where G(x) is the g.f. of A303923.