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%I #10 Oct 13 2020 03:35:30
%S 1,1,2,28,586,16336,559164,22519620,1039209116,53968031108,
%T 3112841732920,197413519635632,13654508980460736,1023144120035225664,
%U 82581014079320743504,7144332294806845079568,659630258631919908187784,64748755209330058463666656,6733915902264715745675338784,739732094650896407811045989408,85594689069528757090534336595600
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} 2^n * ((1+x)^n - A(x))^n.
%C More generally, the following sums are equal:
%C (1) Sum_{n>=0} binomial(n+k-1, n) * r^n * (p + q^n)^n,
%C (2) Sum_{n>=0} binomial(n+k-1, n) * r^n * q^(n^2) / (1 - r*p*q^n)^(n+k),
%C for any fixed integer k; here, k = 1 with r = 2, p = -A(x), q = (1+x).
%H Paul D. Hanna, <a href="/A326282/b326282.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} 2^n * ((1+x)^n - A(x))^n.
%F (2) 1 = Sum_{n>=0} 2^n * (1+x)^(n^2) / (1 + 2*(1+x)^n*A(x))^(n+1).
%F a(n) ~ c * (1 + 2*exp(1/r))^n * r^(2*n) * n! / sqrt(n), where r = 0.925556278640887084941460444526398190071550948416... is the root of the equation exp(1/r) * (1 + 1/(r*LambertW(-exp(-1/r)/r))) = -1/2 and c = 0.06492129653731... - _Vaclav Kotesovec_, Oct 13 2020
%e G.f.: A(x) = 1 + x + 2*x^2 + 28*x^3 + 586*x^4 + 16336*x^5 + 559164*x^6 + 22519620*x^7 + 1039209116*x^8 + 53968031108*x^9 + 3112841732920*x^10 + ...
%e such that
%e 1 = 1 + 2*((1+x) - A(x)) + 2^2*((1+x)^2 - A(x))^2 + 2^3*((1+x)^3 - A(x))^3 + 2^4*((1+x)^4 - A(x))^4 + 2^5*((1+x)^5 - A(x))^5 + 2^6*((1+x)^6 - A(x))^6 + 2^7*((1+x)^7 - A(x))^7 + ...
%e Also,
%e 1 = 1/(1 + 2*A(x)) + 2*(1+x)/(1 + 2*(1+x)*A(x))^2 + 2^2*(1+x)^4/(1 + 2*(1+x)^2*A(x))^3 + 2^3*(1+x)^9/(1 + 2*(1+x)^3*A(x))^4 + 2^4*(1+x)^16/(1 + 2*(1+x)^4*A(x))^5 + 2^5*(1+x)^25/(1 + 2*(1+x)^5*A(x))^6 + 2^6*(1+x)^36/(1 + 2*(1+x)^6*A(x))^7 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=0, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, 2^m*((1+x)^m - Ser(A))^m ) )[#A]/2 ); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A303056, A326283, A326284.
%K nonn
%O 0,3
%A _Paul D. Hanna_, Jun 22 2019
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