A355152 - OEIS

%I #6 Jun 22 2022 02:30:18

%S 1,2,6,21,77,310,1308,5710,25605,117232,545719,2575126,12289906,

%T 59218913,287699288,1407690851,6930731921,34311193558,170691635544,

%U 852874403933,4278234997610,21537155141117,108771673373067,550966271295019,2798396004637028,14248670155477872

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

%H Paul D. Hanna, <a href="/A355152/b355152.txt">Table of n, a(n) for n = 0..400</a>

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

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

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

%F a(n) ~ c * d^n / n^(3/2), where d = 5.4057072074645439585749668665909363742748... and c = 0.8797898450003270547125706339934138835467... - _Vaclav Kotesovec_, Jun 22 2022

%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 21*x^3 + 77*x^4 + 310*x^5 + 1308*x^6 + 5710*x^7 + 25605*x^8 + 117232*x^9 + 545719*x^10 + ...

%e where

%e 0 = ... + x^16/(x^(-4) - A(x))^3 + x^9/(x^(-3) + A(x))^2 + x^4/(x^(-2) - A(x)) + x + (1 - A(x)) + x*(x + A(x))^2 + x^4*(x^2 - A(x))^3 + x^9*(x^3 + A(x))^4 + x^16*(x^4 - A(x))^5 + ...

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

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

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

%Y Cf. A355153.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 21 2022