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A357406 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n * x^(2*n+2) * (1 - x^n)^(n+1).

%I A357406 #9 Oct 08 2022 15:16:30
%S A357406 1,0,-1,0,3,-8,9,0,-10,0,24,-24,0,0,15,0,9,-80,90,0,-43,0,57,-80,13,0,
%T A357406 175,-200,15,-120,313,0,-346,0,450,-168,19,-744,830,0,21,-224,-287,0,
%U A357406 405,0,1014,-1968,25,0,2813,-784,-2448,-360,1575,0,2765,-3520,450,-440,31
%N A357406 Coefficients in the power series expansion of A(x) = Sum_{n=-oo..+oo} n * x^(2*n+2) * (1 - x^n)^(n+1).
%H A357406 Paul D. Hanna, <a href="/A357406/b357406.txt">Table of n, a(n) for n = 0..2050</a>
%F A357406 G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies:
%F A357406 (1) A(x) = Sum_{n=-oo..+oo} n * x^(2*n+2) * (1 - x^n)^(n+1),
%F A357406 (2) A(x) = -Sum_{n=-oo..+oo, n<>0} n * (-1)^n * x^((n-1)*(n-2)) / (1 - x^n)^(n-1).
%e A357406 G.f.: 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 - 80*x^23 + 13*x^24 + ...
%e A357406 Related series.
%e A357406 x/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 + ... + A357401(n)*x^n + ...
%o A357406 (PARI) {a(n) = my(A = sum(m=-n\2-1,n\2+1, m * x^(2*m+2) * (1 - x^m +x*O(x^n) )^(m+1)) ); polcoeff(A,n)}
%o A357406 for(n=0, 40, print1(a(n), ", "))
%Y A357406 Cf. A357401, A356774.
%K A357406 sign
%O A357406 0,5
%A A357406 _Paul D. Hanna_, Sep 27 2022