A184366 - OEIS

%I #14 Mar 05 2017 04:54:21

%S 1,-2,0,0,0,0,7,0,0,0,-21,0,0,0,0,44,0,0,0,0,0,-78,0,0,0,0,0,0,125,0,

%T 0,0,0,0,0,0,-187,0,0,0,0,0,0,0,0,266,0,0,0,0,0,0,0,0,0,-364,0,0,0,0,

%U 0,0,0,0,0,0,483,0,0,0,0,0,0,0,0,0,0,0,-625,0,0,0,0,0,0,0,0,0,0,0,0,792,0,0

%N G.f.: eta(x)^3*(1 - x*eta'(x)/eta(x)), where eta(x) is Dedekind's eta(q) function without the q^(1/24) factor.

%H Seiichi Manyama, <a href="/A184366/b184366.txt">Table of n, a(n) for n = 0..1000</a>

%F G.f.: A(x) = Sum_{n>=0} -(-1)^n * (n-2)(n+3)(2n+1)/6 * x^(n(n+1)/2).

%e G.f.: A(x) = 1 - 2*x + 7*x^6 - 21*x^10 + 44*x^15 - 78*x^21 +...

%e A(x) = eta(x)^3*[1 - x*d/dx log(eta(x))] where

%e eta(x)^3 = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 +...+ (-1)^n*(2n+1)*x^(n(n+1)/2) +...

%e 1 - x*d/dx log(eta(x)) = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 6*x^5 + 12*x^6 + 8*x^7 + 15*x^8 +...+ sigma(n)*x^n +...

%o (PARI) {a(n)=polcoeff(sum(m=0,n,-(-1)^m*(m-2)*(m+3)*(2*m+1)/6*x^(m*(m+1)/2)),n)}

%o (PARI) {a(n)=polcoeff(eta(x+x*O(x^n))^3*(1-x*deriv(log(eta(x+x*O(x^n))))),n)}

%Y Cf. A000203, A184363, A184365, A283334.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jan 17 2011