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%I #17 Sep 17 2023 21:31:02
%S 1,2,5,27,143,833,5198,33607,223627,1522249,10546221,74119591,
%T 527150783,3786896705,27437431852,200267244944,1471209231873,
%U 10869315344076,80707738490984,601977204069443,4508156389422426,33884634730883602,255532279985062648,1932864141175160374,14660843479381675987,111486308441258038306,849773662058395948696,6491244696415245552638,49685280480631490670702,381014689125058139363522,2926949265189880054761750
%N G.f. A(x) satisfies: A(x) = Sum_{n=-oo..+oo} x^n * (A(x) - x^n)^(2*n).
%C Compare to: Sum_{n=-oo..+oo} x^n * (c - x^n)^n = 0 for fixed |c| > 0.
%H Paul D. Hanna, <a href="/A265266/b265266.txt">Table of n, a(n) for n = 0..200</a>
%F The g.f. A(x) = Sum_{n>=0} a(n)*x^n also satisfies:
%F (1) A(x) = Sum_{n=-oo..+oo} x^n * (A(x) + x^n)^(2*n).
%F (2) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 - x^n*A(x))^(2*n).
%F (3) A(x) = Sum_{n=-oo..+oo} x^(2*n^2-n) / (1 + x^n*A(x))^(2*n).
%F (4) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-x)^n * (A(x^2) - x^n)^n.
%F (5) A(x^2) = (1/2) * Sum_{n=-oo..+oo} (-x)^n * (A(x^2) + x^n)^n.
%F (6) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 - x^n*A(x^2))^n.
%F (7) A(x^2) = (1/2) * Sum_{n=-oo..+oo} x^(n^2-n) / (1 + x^n*A(x^2))^n.
%F a(n) ~ c * d^n / n^(3/2), where d = 8.078575206447883305059904... and c = 0.294232997886629805825... - _Vaclav Kotesovec_, Sep 03 2017
%e G.f.: A(x) = 1 + 2*x + 5*x^2 + 27*x^3 + 143*x^4 + 833*x^5 + 5198*x^6 + 33607*x^7 + 223627*x^8 + 1522249*x^9 + 10546221*x^10 + ...
%e Let A = g.f. A(x) where A(x) = P(x) + N(x) then
%e P(x) = 1 + x*(A - x)^2 + x^2*(A - x^2)^4 + x^3*(A - x^3)^6 + x^4*(A - x^4)^8 + x^5*(A - x^5)^10 + x^6*(A - x^6)^12 + x^7*(A - x^7)^14 + x^8*(A - x^8)^16 + ...
%e N(x) = x/(1-x*A)^2 + x^6/(1-x^2*A)^4 + x^15/(1-x^3*A)^6 + x^28/(1-x^4*A)^8 + x^45/(1-x^5*A)^10 + x^66/(1-x^6*A)^12 + x^91/(1-x^7*A)^14 + ...
%e Explicitly,
%e P(x) = 1 + x + 3*x^2 + 20*x^3 + 117*x^4 + 708*x^5 + 4535*x^6 + 29801*x^7 + 200369*x^8 + 1373999*x^9 + 9570641*x^10 + 67539460*x^11 + 481899317*x^12 + ...
%e N(x) = x + 2*x^2 + 7*x^3 + 26*x^4 + 125*x^5 + 663*x^6 + 3806*x^7 + 23258*x^8 + 148250*x^9 + 975580*x^10 + 6580131*x^11 + 45251466*x^12 + ...
%o (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A=Vec( sum(n=-#A-1,#A+1, x^n*(Ser(A) - x^n)^(2*n) ) ) );A[n+1]}
%o for(n=0,40,print1(a(n),", "))
%o (PARI) /* Quick print of terms 0..N (informal): */
%o N = 40; A=[1]; for(i=1,N, A=Vec( sum(n=-#A-1,#A+1, x^n*(Ser(A) - x^n)^(2*n) ) ) );A
%Y Cf. A260147.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jan 03 2016
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