%I #6 Jun 22 2022 02:43:10

%S 1,3,10,47,248,1354,7832,46672,285136,1775728,11232028,71959338,

%T 465981585,3045116666,20055877798,132995908915,887220714352,

%U 5950074234287,40092205226021,271289342487845,1842724189708458,12559944025175681,85877738644538351

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

%H Paul D. Hanna, <a href="/A355154/b355154.txt">Table of n, a(n) for n = 1..400</a>

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

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

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

%F (3) 1/(1+x) = Sum_{n=-oo..+oo} (-1)^n * (x + x^2)^(n^2) * (1+x)^(2*n) / (1 - A(x)*(1+x)^(n+1))^(n+1).

%F a(n) ~ c * d^n / n^(3/2), where d = 7.312110954492511257173117... and c = 0.1256708322688258093501... - _Vaclav Kotesovec_, Jun 22 2022

%e G.f.: A(x) = x + 3*x^2 + 10*x^3 + 47*x^4 + 248*x^5 + 1354*x^6 + 7832*x^7 + 46672*x^8 + 285136*x^9 + 1775728*x^10 + 11232028*x^11 + ...

%e where

%e 1 = ... + x^16/(1/(1+x)^5 - A(x))^5 - x^9/(1/(1+x)^4 - A(x))^4 + x^4/(1/(1+x)^3 - A(x))^3 - x/(1/(1+x)^2 - A(x))^2 + 1/(1/(1+x) - A(x)) - x + x^4*((1+x) - A(x)) - x^9*((1+x)^2 - A(x))^2 + x^16*((1+x)^3 - A(x))^3 -+ ...

%o (PARI) {a(n) = my(A=[0,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(n+4));

%o A[#A] = -polcoeff( sum(n=-t,t, (-x)^((n+1)^2) * ((1+x)^n - Ser(A))^n ), #A-1));A[n+1]}

%o for(n=1,30,print1(a(n),", "))

%K nonn

%O 1,2

%A _Paul D. Hanna_, Jun 21 2022