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

%I #6 Jun 22 2022 02:55:08

%S 1,1,7,28,184,1024,6676,42367,282765,1897203,13004369,89991470,

%T 630242521,4450613382,31683411117,227041605009,1636747514265,

%U 11860982110191,86356914006201,631382010617369,4633749074928932,34124201919637479,252086975581304199

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

%H Paul D. Hanna, <a href="/A355155/b355155.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+1) - 2*A(x))^n.

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

%F a(n) ~ c * d^n / n^(3/2), where d = 7.89889... and c = 0.06269... - _Vaclav Kotesovec_, Jun 22 2022

%e G.f.: A(x) = x + x^2 + 7*x^3 + 28*x^4 + 184*x^5 + 1024*x^6 + 6676*x^7 + 42367*x^8 + 282765*x^9 + 1897203*x^10 + 13004369*x^11 + ...

%e where

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

%o (PARI) {a(n) = my(A=[0,1],t); for(i=1,n, A=concat(A,0); t = ceil(sqrt(2*n+4));

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

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

%Y Cf. A355156.

%K nonn

%O 1,3

%A _Paul D. Hanna_, Jun 21 2022