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%I #17 Aug 13 2018 03:55:47
%S 1,4,26,356,8871,320672,14811200,820185072,52546341422,3808527303300,
%T 307523461730866,27352330591164308,2656394433081980649,
%U 279696497208771609120,31739466678890197201328,3862114024795578127697248,501700135604304149492422266,69305144023051764776753873168,10145743117833906529065611237208,1569100081969097895595627120200512
%N G.f. A(x) satisfies: Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n) )^n = 1.
%H Paul D. Hanna, <a href="/A317668/b317668.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n) )^n.
%F (2) A(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n.
%F (3) 1 = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(4*n+4).
%F (4) Let B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+1) )^n,
%F then B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(3*n+3).
%F (5) Let C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+2) )^n,
%F then C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(2*n+2).
%F (6) Let D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+3) )^n,
%F then D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(n+1).
%F a(n) ~ 2^(2*n + log(2)/8 - 5/2) * n^n / (sqrt(1-log(2)) * exp(n) * (log(2))^(2*n+1)). - _Vaclav Kotesovec_, Aug 13 2018
%e G.f.: A(x) = 1 + 4*x + 26*x^2 + 356*x^3 + 8871*x^4 + 320672*x^5 + 14811200*x^6 + 820185072*x^7 + 52546341422*x^8 + 3808527303300*x^9 + 307523461730866*x^10 + ...
%e such that
%e 1 = 1 + (1/A(x) - (1-x)^4) + (1/A(x) - (1-x)^8)^2 + (1/A(x) - (1-x)^12)^3 + (1/A(x) - (1-x)^16)^4 + (1/A(x) - (1-x)^20)^5 + (1/A(x) - (1-x)^24)^6 + (1/A(x) - (1-x)^28)^7 + (1/A(x) - (1-x)^32)^8 + ...
%e Also,
%e A(x) = 1 + (1/A(x) - (1-x)^8) + (1/A(x) - (1-x)^12)^2 + (1/A(x) - (1-x)^16)^3 + (1/A(x) - (1-x)^20)^4 + (1/A(x) - (1-x)^24)^5 + (1/A(x) - (1-x)^28)^6 + (1/A(x) - (1-x)^32)^7 + (1/A(x) - (1-x)^36)^8 + ...
%e RELATED SERIES.
%e (1) The series B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+1) )^n begins
%e B(x) = 1 + x + 5*x^2 + 67*x^3 + 1669*x^4 + 60246*x^5 + 2781335*x^6 + 154062232*x^7 + 9875799121*x^8 + 716231200582*x^9 + 57865799711347*x^10 + ...
%e also given by B(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(3*n+3).
%e (2) The series C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+2) )^n begins
%e C(x) = 1 + 2*x + 11*x^2 + 148*x^3 + 3683*x^4 + 132888*x^5 + 6131332*x^6 + 339397944*x^7 + 21742672693*x^8 + 1575995237188*x^9 + 127268039660042*x^10 + ...
%e also given by C(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(2*n+2).
%e (3) The series D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+3) )^n begins
%e D(x) = 1 + 3*x + 18*x^2 + 244*x^3 + 6073*x^4 + 219238*x^5 + 10117351*x^6 + 560000464*x^7 + 35868610134*x^8 + 2599382401532*x^9 + 209871544727484*x^10 + ...
%e also given by D(x) = Sum_{n>=0} ( 1/A(x) - (1-x)^(4*n+4) )^n * (1-x)^(n+1).
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0); A[#A] = Vec( sum(m=0, #A, ( 1/Ser(A) - (1-x)^(4*m+4) )^m ) )[#A]/2 ); A[n+1]}
%o for(n=0, 25, print1(a(n), ", "))
%Y Cf. A317349, A317666, A317667, A317803.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Aug 12 2018
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