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%I #38 Sep 29 2023 10:30:15
%S 1,1,1,8,89,1326,24247,521764,12867985,357229785,11017306489,
%T 373675921093,13825260663882,554216064798423,23934356706763264,
%U 1108017262467214486,54747529760516714323,2876096694574711401525,160092696678371426933342,9413031424290635395882462,583000844360279565483710624
%N G.f. A(x) satisfies: 1 = Sum_{n>=0} ((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 = 1, p = -A(x), q = (1+x). - _Paul D. Hanna_, Jun 22 2019
%H Paul D. Hanna, <a href="/A303056/b303056.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) 1 = Sum_{n>=0} ((1+x)^n - A(x))^n.
%F (2) 1 = Sum_{n>=0} (1+x)^(n^2) / (1 + (1+x)^n*A(x))^(n+1).
%F a(n) ~ c * d^n * n! / sqrt(n), where d = A317855 = 3.1610886538654... and c = 0.11739505492506... - _Vaclav Kotesovec_, Sep 26 2020
%e G.f.: A(x) = 1 + x + x^2 + 8*x^3 + 89*x^4 + 1326*x^5 + 24247*x^6 + 521764*x^7 + 12867985*x^8 + 357229785*x^9 + 11017306489*x^10 + ...
%e such that
%e 1 = 1 + ((1+x) - A(x)) + ((1+x)^2 - A(x))^2 + ((1+x)^3 - A(x))^3 + ((1+x)^4 - A(x))^4 + ((1+x)^5 - A(x))^5 + ((1+x)^6 - A(x))^6 + ((1+x)^7 - A(x))^7 + ...
%e Also,
%e 1 = 1/(1 + A(x)) + (1+x)/(1 + (1+x)*A(x))^2 + (1+x)^4/(1 + (1+x)^2*A(x))^3 + (1+x)^9/(1 + (1+x)^3*A(x))^4 + (1+x)^16/(1 + (1+x)^4*A(x))^5 + (1+x)^25/(1 + (1+x)^5*A(x))^6 + (1+x)^36/(1 + (1+x)^6*A(x))^7 + ...
%e RELATED SERIES.
%e log(A(x)) = x + x^2/2 + 22*x^3/3 + 325*x^4/4 + 6186*x^5/5 + 137380*x^6/6 + 3478651*x^7/7 + 98674253*x^8/8 + 3096911434*x^9/9 + ...
%e PARTICULAR VALUES.
%e Although the power series A(x) diverges at x = -1/2, it may be evaluated formally.
%e Let t = A(-1/2) = 0.545218973635949431234950245034944106957612798888179456724264...
%e then t satisfies
%e (1) 1 = Sum_{n>=0} ( 1/2^n - t )^n.
%e (2) 1 = Sum_{n>=0} 2^n / ( 2^n + t )^(n+1).
%e Also,
%e A(r) = 1/2 at r = -0.54683649902292991492196620520872286547799291909992048564578...
%e where
%e (1) 1 = Sum_{n>=0} ( (1+r)^n - 1/2 )^n.
%e (2) 1 = Sum_{n>=0} (1+r)^(-n) / ( 1/(1+r)^n + 1/2 )^(n+1).
%o (PARI) {a(n) = my(A=[1]); for(i=0,n, A=concat(A,0); A[#A] = Vec( sum(m=0,#A, ((1+x)^m - Ser(A))^m ) )[#A] );A[n+1]}
%o for(n=0,30, print1(a(n),", "))
%Y Cf. A304642, A304639, A303926.
%Y Cf. A321602, A321603, A321604, A321605.
%Y Cf. A326282, A326283, A326284.
%Y Cf. A337755, A337756, A337757.
%K nonn
%O 0,4
%A _Paul D. Hanna_, Apr 19 2018
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