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G.f. A(x) satisfies: 0 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * (x^n - (-1)^n*2*A(x))^(n+1).
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%I #6 Jun 22 2022 02:25:03

%S 1,2,10,49,272,1617,10082,65101,431635,2921557,20104828,140240820,

%T 989366180,7046832503,50604822586,365995915453,2663552184585,

%U 19490777248544,143322830835474,1058514890796268,7848460747315854,58400364116985559,435963301942052908

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

%H Paul D. Hanna, <a href="/A355153/b355153.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*(n+1)/2) * (x^n - (-1)^n*2*A(x))^(n+1).

%F (2) 0 = Sum_{n=-oo..+oo} x^(3*n*(n-1)/2) / (1 - (-x)^n*2*A(x))^(n-1).

%F a(n) ~ c * d^n / n^(3/2), where d = 7.9902080716257105993518109688307894114113... and c = 0.651281200395360554284404869149552977212. - _Vaclav Kotesovec_, Jun 22 2022

%e G.f.: A(x) = 1 + 2*x + 10*x^2 + 49*x^3 + 272*x^4 + 1617*x^5 + 10082*x^6 + 65101*x^7 + 431635*x^8 + 2921557*x^9 + 20104828*x^10 + ...

%e where

%e 0 = ... + x^6/(x^(-4) - 2*A(x))^3 + x^3/(x^(-3) + 2*A(x))^2 + x/(x^(-2) - 2*A(x)) + 1 + (1 - 2*A(x)) + x*(x + 2*A(x))^2 + x^3*(x^2 - 2*A(x))^3 + x^6*(x^3 + 2*A(x))^4 +--+ ...

%o (PARI) {a(n) = my(A=[1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+1));

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

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

%Y Cf. A355152.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Jun 21 2022