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%I #8 Jun 23 2018 22:51:37
%S -1,1,-1,-1,5,-8,-1,30,-54,-5,179,-178,-608,1518,2611,-18198,24294,
%T 73472,-365936,454378,1238770,-6059472,7396293,18989602,-90050185,
%U 97765822,309199646,-1270466588,950395782,5591065880,-17812121981,3192879156,107153213918,-255721287314,-142492705829,2046576827997,-3763046892266,-5107717145521,37668440369581,-55097197661332,-122428866718927
%N G.f. A(x) satisfies: Sum_{n>=0} Product_{k=1..n} x^(n+1-k) + A(x)^k = 1.
%H Paul D. Hanna, <a href="/A306068/b306068.txt">Table of n, a(n) for n = 1..100</a>
%F G.f. A(x) satisfies: A(A(x)) = x.
%e G.f.: A(x) = -x + x^2 - x^3 - x^4 + 5*x^5 - 8*x^6 - x^7 + 30*x^8 - 54*x^9 - 5*x^10 + 179*x^11 - 178*x^12 - 608*x^13 + 1518*x^14 + 2611*x^15 - 18198*x^16 + ...
%e such that
%e 1 = 1 + (x + A(x)) + (x + A(x)^2)*(x^2 + A(x)) + (x + A(x)^3)*(x^2 + A(x)^2)*(x^3 + A(x)) + (x + A(x)^4)*(x^2 + A(x)^3)*(x^3 + A(x)^2)*(x^4 + A(x)) + (x + A(x)^5)*(x^2 + A(x)^4)*(x^3 + A(x)^3)*(x^4 + A(x)^2)*(x^5 + A(x)) + ...
%e also, A(A(x)) = x.
%o (PARI) {a(n) = my(A=[-1]); for(i=1, n, A = concat(A, 0); A[#A] = -Vec( sum(n=0, #A, prod(k=1, n, x^(n+1-k) + (x*Ser(A))^k ) ) )[#A+1] ); A[n]}
%o for(n=1, 30, print1(a(n), ", "))
%Y Cf. A306088, A306089.
%K sign
%O 1,5
%A _Paul D. Hanna_, Jun 23 2018
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