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%I A357232 #12 Dec 03 2022 12:04:53
%S A357232 1,3,25,254,2763,32180,393169,4964017,64254694,848214039,11375359344,
%T A357232 154547261539,2122630191360,29423373611509,411105855956011,
%U A357232 5783709944279141,81862107418919278,1164873718427628846,16654829725736560441,239140138388082634266,3446933945466334214525
%N A357232 a(n) = coefficient of x^n, n >= 0, in A(x) such that: 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^n)^(2*n+1).
%C A357232 Compare to A355865.
%H A357232 Paul D. Hanna, <a href="/A357232/b357232.txt">Table of n, a(n) for n = 0..300</a>
%F A357232 Generating function A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
%F A357232 (1) 2 = Sum_{n=-oo..+oo} (-1)^n * x^n * (2*A(x) + x^n)^(2*n+1).
%F A357232 (2) 2 = Sum_{n=-oo..+oo} (-1)^n * x^(2*n*(n-1)) / (1 + 2*A(x)*x^n)^(2*n-1).
%e A357232 G.f.: A(x) = 1 + 3*x + 25*x^2 + 254*x^3 + 2763*x^4 + 32180*x^5 + 393169*x^6 + 4964017*x^7 + 64254694*x^8 + 848214039*x^9 + 11375359344*x^10 + ...
%o A357232 (PARI) {a(n) = my(A=1); for(L=1,n, A = truncate(A);
%o A357232 A = A + 1 - (1/2)*sum(m=-L,L, (-1)^m * x^m * (2*A + x^m +x^2*O(x^(L+1)))^(2*m+1) ) ); polcoeff(A,n)}
%o A357232 for(n=0,30, print1(a(n),", "))
%Y A357232 Cf. A355865, A355868, A357402.
%K A357232 nonn
%O A357232 0,2
%A A357232 _Paul D. Hanna_, Oct 14 2022

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