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A357401 Coefficients in the power series expansion of 1/Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).

%I A357401 #16 Sep 29 2022 17:46:40
%S A357401 1,0,1,0,-2,8,-14,16,-7,-24,103,-232,334,-256,-211,1400,-3562,6048,
%T A357401 -6470,512,17788,-53720,102983,-134832,76147,187960,-776169,1690880,
%U A357401 -2558499,2270952,1214672,-10443024,26674201,-45822896,51953043,-11147384,-126256811,401311496
%N A357401 Coefficients in the power series expansion of 1/Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).
%C A357401 Equals column 1 of triangle A357400. The g.f. G(x,y) of triangle A357400 satisfies: y = Sum_{n=-oo..+oo} x^(2*n+1) * (1 - x^n)^(n+1) * G(x,y)^n; this sequence gives the coefficients of x^n*y in G(x,y) for n >= 1.
%H A357401 Paul D. Hanna, <a href="/A357401/b357401.txt">Table of n, a(n) for n = 1..1500</a>
%F A357401 G.f. A(x,y) = Sum_{n>=1} a(n) * x^n satisfies the following relations.
%F A357401 (1) A(x) = 1/Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).
%F A357401 (2) A(X) = -x/Sum_{n=-oo..+oo} n * (-1)^n * x^(n*(n-1)) / (1 - x^(n+1))^n.
%e A357401 G.f.: A(x) = x + x^3 - 2*x^5 + 8*x^6 - 14*x^7 + 16*x^8 - 7*x^9 - 24*x^10 + 103*x^11 - 232*x^12 + 334*x^13 - 256*x^14 - 211*x^15 + 1400*x^16 - 3562*x^17 + 6048*x^18 - 6470*x^19 + 512*x^20 + ...
%e A357401 Related series.
%e A357401 x/A(x) = 1 - x^2 + 3*x^4 - 8*x^5 + 9*x^6 - 10*x^8 + 24*x^10 - 24*x^11 + 15*x^14 + 9*x^16 - 80*x^17 + 90*x^18 - 43*x^20 + 57*x^22 + ... + A357406(n)*x^n + ...
%e A357401 which equals x*Sum_{n=-oo..+oo} n * x^(2*n+1) * (1 - x^n)^(n+1).
%o A357401 (PARI) {a(n) = my(A = 1/sum(m=-n-1,n+1, m * x^(2*m+1) * (1 - x^m +x*O(x^n) )^(m+1) )); polcoeff(A, n)}
%o A357401 for(n=1,40, print1(a(n), ", "))
%o A357401 (PARI) /* As Column 1 of triangle A357400 (slow) */
%o A357401 {A357400(n,k) = my(A=[1]); for(i=0, n, A = concat(A, 0);
%o A357401 A[#A] = polcoeff(y - sum(m=-#A\2-1, #A\2+1, x^(2*m+1) * (1 - x^m +x*O(x^#A))^(m+1) * Ser(A)^m  ), #A-2); ); polcoeff(A[n+1],k,y)}
%o A357401 for(n=1, 40, print1(A357400(n,1), ", "))
%Y A357401 Cf. A357400, A357406.
%K A357401 sign
%O A357401 1,5
%A A357401 _Paul D. Hanna_, Sep 26 2022