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%I #10 Mar 05 2023 12:07:09
%S 1,4,32,400,6016,99968,1779456,33343488,650141696,13084840960,
%T 270257033216,5704378748928,122667151491072,2681371746680832,
%U 59480466149277696,1337376871507230720,30452122562013954048,701788662787645112320,16362889157705834954752,385927625558617880526848
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} (-x)^n * (A(x)^n + (-1)^n)^n.
%C a(n) = 2^n * A360949(n) for n >= 0.
%H Paul D. Hanna, <a href="/A325574/b325574.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} (-x)^n * (A(x)^n + (-1)^n)^n.
%F (2) 1 = Sum_{n>=0} (-x)^n * A(x)^(n^2) / (1 - (-1)^n*x*A(x)^n)^(n+1).
%e G.f.: A(x) = 1 + 4*x + 32*x^2 + 400*x^3 + 6016*x^4 + 99968*x^5 + 1779456*x^6 + 33343488*x^7 + 650141696*x^8 + 13084840960*x^9 + 270257033216*x^10 + ...
%e such that
%e 1 = 1 - x*(A(x) - 1) + x^2*(A(x)^2 + 1)^2 - x^3*(A(x)^3 - 1)^3 + x^4*(A(x)^4 + 1)^4 - x^5*(A(x)^5 - 1)^5 + x^6*(A(x)^6 + 1)^6 - x^7*(A(x)^7 - 1)^7 + x^8*(A(x)^8 + 1)^8 - x^9*(A(x)^9 - 1)^9 + x^10*(A(x)^10 + 1)^10 + ...
%e also,
%e 1 = 1/(1 - x) - x*A(x)/(1 + x*A(x))^2 + x^2*A(x)^4/(1 - x*A(x)^2)^3 - x^3*A(x)^9/(1 + x*A(x)^3)^4 + x^4*A(x)^16/(1 - x*A(x)^4)^5 - x^5*A(x)^25/(1 + x*A(x)^5)^6 + x^6*A(x)^36/(1 - x*A(x)^6)^7 - x^7*A(x)^49/(1 + x*A(x)^7)^8 + x^8*A(x)^64/(1 - x*A(x)^8)^9 - x^9*A(x)^81/(1 + x*A(x)^9)^10 + ...
%e RELATED SERIES.
%e A(x/2) is an integer series:
%e A(x/2) = 1 + 2*x + 8*x^2 + 50*x^3 + 376*x^4 + 3124*x^5 + 27804*x^6 + 260496*x^7 + 2539616*x^8 + 25556330*x^9 + ... + A360949(n)*x^n + ...
%o (PARI) {a(n) = my(A=[1]);
%o for(i=1,n, A=concat(A,0); A[#A] = polcoeff(sum(m=0,#A, (-x)^m*(Ser(A)^m + (-1)^m)^m ),#A));A[n+1]}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A360949.
%K nonn
%O 0,2
%A _Paul D. Hanna_, May 16 2019
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