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%I #6 Aug 11 2022 07:23:50
%S 1,1,3,8,26,69,186,554,1606,4772,14825,45920,143586,455561,1446777,
%T 4622609,14885492,48099814,156175165,509527438,1667502213,5475430124,
%U 18036181574,59562083008,197197284543,654439115129,2176448986396,7252784407286,24214655260997
%N G.f. A(x) satisfies: x = Sum_{n=-oo..+oo} (-x)^n * (1 + A(x)*x^n)^n.
%C Related identity: Sum_{n=-oo..+oo} (-x)^n * (x^n + y)^n = 0 for all y.
%C Related identity: Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (1 + y*x^n)^n = 0 for all y.
%H Paul D. Hanna, <a href="/A355869/b355869.txt">Table of n, a(n) for n = 0..200</a>
%F G.f. A(x) satisfies:
%F (1) x = Sum_{n=-oo..+oo} (-x)^n * (1 + A(x)*x^n)^n.
%F (2) 0 = Sum_{n=-oo..+oo} (-x)^n * (1 + A(x)*x^n)^n / A(x)^n.
%F (3) x = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) / (x^n + A(x))^n.
%F (4) 0 = Sum_{n=-oo..+oo} (-1)^n * x^(n*(n-1)) * A(x)^n / (x^n + A(x))^n.
%F (5) 0 = Sum_{n=-oo..+oo} (-x)^n * (1 + A(x)^(n+1)*x^n)^n.
%e G.f.: A(x) = 1 + x + 3*x^2 + 8*x^3 + 26*x^4 + 69*x^5 + 186*x^6 + 554*x^7 + 1606*x^8 + 4772*x^9 + 14825*x^10 + 45920*x^11 + 143586*x^12 + ...
%e where
%e x = ... - x^(-3)/(1 + A(x)*x^(-3))^3 + x^(-2)/(1 + A(x)*x^(-2))^2 - x^(-1)/(1 + A(x)*x^(-1)) + 1 - x*(1 + A(x)*x) + x^2*(1 + A(x)*x^2)^2 - x^3*(1 + A(x)*x^3)^3 + ... + (-x)^n*(1 + A(x)*x^n)^n + ...
%o (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A=concat(A,0);
%o A[#A] = -polcoeff( sum(m=-#A,#A, (-x)^m*(1 + Ser(A)*x^m)^m ), #A-1));A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%o (PARI) {a(n) = my(A=[1,1]); for(i=1,n, A=concat(A,0);
%o A[#A] = -polcoeff( sum(m=-#A,#A, (-1)^m*x^(m*(m-1))/(x^m + Ser(A))^m ), #A-1));A[n+1]}
%o for(n=0,30,print1(a(n),", "))
%K nonn
%O 0,3
%A _Paul D. Hanna_, Aug 09 2022
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