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%I #9 May 16 2023 04:22:05
%S 1,1,6,51,470,4716,49350,534115,5929892,67175779,773473709,9025907984,
%T 106511693025,1268898400188,15240421643846,184348620664449,
%U 2243749948233175,27459089491691552,337685454820968084,4170918486201555250,51719670553572755173,643610071084847351183
%N Expansion of g.f. A(x) satisfying 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(2*n-1).
%H Paul D. Hanna, <a href="/A363112/b363112.txt">Table of n, a(n) for n = 0..300</a>
%F Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F (1) 1 = Sum_{n=-oo..+oo} x^n * (2*A(x) - x^n)^(2*n-1).
%F (2) -1 = Sum_{n=-oo..+oo} x^(2*n^2) / (1 - 2*A(x)*x^n)^(2*n+1).
%e G.f.: A(x) = 1 + x + 6*x^2 + 51*x^3 + 470*x^4 + 4716*x^5 + 49350*x^6 + 534115*x^7 + 5929892*x^8 + 67175779*x^9 + 773473709*x^10 + ...
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A[#A] = polcoeff(-1 + sum(m=-#A, #A, x^m * (2*Ser(A) - x^m)^(2*m-1) ), #A-1)/2); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%o (PARI) {a(n) = my(A=[1]); for(i=1, n, A=concat(A, 0);
%o A[#A] = polcoeff(1 - sum(m=-#A, #A, x^(2*m^2)/(1 - 2*Ser(A)*x^m)^(2*m+1) ), #A-1)/2); A[n+1]}
%o for(n=0, 30, print1(a(n), ", "))
%Y Cf. A357227, A363113, A363114.
%Y Cf. A361772.
%K nonn
%O 0,3
%A _Paul D. Hanna_, May 14 2023
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