A303925 - OEIS

%I #11 Dec 13 2018 23:26:55

%S 1,1,3,12,56,288,1587,9222,55957,352267,2290842,15343839,105634437,

%T 746478622,5409932286,40189454704,305972524737,2387238374532,

%U 19090018863000,156496468777604,1315509548959765,11341506519584442,100300906407392783,909967403153604712,8468614126450656268,80832677102193209308,791071858022525348235

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

%H Paul D. Hanna, <a href="/A303925/b303925.txt">Table of n, a(n) for n = 0..300</a>

%F G.f. A(x) satisfies:

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

%F (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

%F 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.

%F 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.

%e 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 + ...

%e such that

%e 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 + ...

%o (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]}

%o for(n=0,30, print1(a(n),", "))

%Y Cf. A303923, A303924.

%K nonn

%O 0,3

%A _Paul D. Hanna_, May 03 2018