A184363 - OEIS

%I #6 Mar 30 2012 18:37:25

%S 1,-4,0,10,0,0,-21,0,0,0,39,0,0,0,0,-66,0,0,0,0,0,104,0,0,0,0,0,0,

%T -155,0,0,0,0,0,0,0,221,0,0,0,0,0,0,0,0,-304,0,0,0,0,0,0,0,0,0,406,0,

%U 0,0,0,0,0,0,0,0,0,-529,0,0,0,0,0,0,0,0,0,0,0,675,0,0,0,0,0,0,0,0,0,0,0,0,-846

%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.

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

%F G.f.: A(x) = eta(x)^2*G(x) where G(x) is the g.f. of A184362.

%e G.f.: A(x) = 1 - 4*x + 10*x^3 - 21*x^6 + 39*x^10 - 66*x^15 +...

%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*(2*m+1)*(m^2+m+6)/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. A184362, A184366.

%K sign

%O 0,2

%A _Paul D. Hanna_, Jan 18 2011