%I #9 Jul 30 2022 15:38:12
%S 1,2,6,25,112,557,2914,15837,88531,505581,2936676,17294352,103018292,
%T 619595991,3757342674,22948207189,141033508661,871527612640,
%U 5412015056754,33754524947592,211353845133650,1328099943458743,8372466442163468,52936608451071755
%N G.f. A(x) satisfies: 0 = Sum_{n=-oo..+oo} x^(n*(n+1)/2) * (x^n - 2*A(x))^(n+1).
%H Paul D. Hanna, <a href="/A355862/b355862.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 - 2*A(x))^(n+1).
%F (2) 0 = Sum_{n=-oo..+oo} x^(n*(3*n-1)/2) / (1 - 2*A(x)*x^n)^(n-1).
%F a(n) ~ c * d^n / n^(3/2), where d = 6.74709799536602052858389740164829219437... and c = 0.517304287814827280375970612560243586... - _Vaclav Kotesovec_, Jul 23 2022
%F A(1/d) = 2.022729610323037319... where 1/d = 0.148211868374642... and d is the value given above by _Vaclav Kotesovec_. - _Paul D. Hanna_, Jul 30 2022
%e G.f.: A(x) = 1 + 2*x + 6*x^2 + 25*x^3 + 112*x^4 + 557*x^5 + 2914*x^6 + 15837*x^7 + 88531*x^8 + 505581*x^9 + 2936676*x^10 + 17294352*x^11 + ...
%e where
%e 0 = ... + x^6/(1/x^4 - 2*A(x))^3+ x^3/(1/x^3 - 2*A(x))^2 + x/(1/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 - A(x))^4 + ... + x^(n*(n+1)/2)*(x^n - 2*A(x))^(n+1) + ...
%e Specific values.
%e A(0.148188601...) = 2.
%e A(1/7) = 1.72240285856328...
%o (PARI) {a(n) = my(A=[1],M); for(i=1,n, A=concat(A,0); M = ceil(sqrt(2*(#A)+9));
%o A[#A] = polcoeff( sum(m=-M,M, x^(m*(m+1)/2) * (x^m - 2*Ser(A))^(m+1) ), #A-1)/2);A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jul 22 2022
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