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%I #6 Jun 22 2022 02:56:28
%S 1,3,7,34,156,635,2966,14327,66047,314700,1560159,7710390,38251395,
%T 192549444,974601369,4952273462,25349634715,130414412847,673028519274,
%U 3487306501970,18139905710007,94640387701755,495160150021161,2597800048395361,13661957722514619
%N G.f. A(x) satisfies: 2 = Sum_{n=-oo..+oo} (-x)^(n*(n+1)/2) * ((1+x)^n + 2*A(x))^(n+1).
%H Paul D. Hanna, <a href="/A355156/b355156.txt">Table of n, a(n) for n = 1..400</a>
%F G.f. A(x) satisfies:
%F (1) 2 = Sum_{n=-oo..+oo} (-x)^(n*(n+1)/2) * ((1+x)^n + 2*A(x))^(n+1).
%F (2) 2 = Sum_{n=-oo..+oo} (-x)^(n*(n+1)/2) * (1+x)^(n*(n+1)) / (1 + 2*A(x)*(1+x)^(n+1))^n.
%F a(n) ~ c * d^n / n^(3/2), where d = 5.5806... and c = 0.3849... - _Vaclav Kotesovec_, Jun 22 2022
%e G.f.: A(x) = x + 3*x^2 + 7*x^3 + 34*x^4 + 156*x^5 + 635*x^6 + 2966*x^7 + 14327*x^8 + 66047*x^9 + 314700*x^10 + 1560159*x^11 + ...
%e where
%e 2 = ... + x^6/(1/(1+x)^4 + 2*A(x))^3 - x^3/(1/(1+x)^3 + 2*A(x))^2 - x/(1/(1+x)^2 + 2*A(x)) + 1 + (1 + 2*A(x)) - x*((1+x) + 2*A(x))^2 - x^3*((1+x)^2 + 2*A(x))^3 + x^6*((1+x)^3 + 2*A(x))^4 +--+ ...
%o (PARI) {a(n) = my(A=[0,1],t); for(i=1,n, A=concat(A,0); t=ceil(sqrt(2*n+9));
%o A[#A] = -polcoeff( sum(n=-t,t, (-x)^(n*(n+1)/2) * ((1+x)^n + 2*Ser(A))^(n+1) ), #A-1)/2);A[n+1]}
%o for(n=1,30,print1(a(n),", "))
%Y Cf. A355155.
%K nonn
%O 1,2
%A _Paul D. Hanna_, Jun 21 2022
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