%I #9 Aug 22 2024 16:47:33
%S 1,3,9,32,111,378,1287,4395,15012,51247,174930,597177,2038676,6959625,
%T 23758677,81107291,276883938,945225504,3226807479,11015664750,
%U 37605240819,128376648392,438251781660,1496102499171,5107389823160,17435590684584,59521562482293
%N G.f. 1/[Sum_{n>=0} (2*n+1)*(-x)^(n*(n+1)/2)].
%C The logarithmic derivative equals 3 times A202144.
%C Radius of convergence r is approximately equal to:
%C r = 0.29292898163912377571341042979083759105819894028205070...
%C where limit a(n)*r^n = 0.81375788478450306387071671675851320912210...
%e G.f.: A(x) = 1 + 3*x + 9*x^2 + 32*x^3 + 111*x^4 + 378*x^5 + 1287*x^6 +...
%e where 1/A(x) = 1 - 3*x - 5*x^3 + 7*x^6 + 9*x^10 - 11*x^15 - 13*x^21 + 15*x^28 + 17*x^36 +...+ (2*n+1)*(-x)^(n*(n+1)/2) +...
%e Compare g.f. A(x) to the cube of the partition function:
%e P(x)^3 = 1 + 3*x + 9*x^2 + 22*x^3 + 51*x^4 + 108*x^5 + 221*x^6 +...
%e where 1/P(x)^3 = 1 - 3*x + 5*x^3 - 7*x^6 + 9*x^10 - 11*x^15 + 13*x^21 - 15*x^28 + 17*x^36 +...+ (-1)^n*(2*n+1)*x^(n*(n+1)/2) +...
%o (PARI) {a(n)=polcoeff(1/sum(k=0,sqrtint(2*n+1),(2*k+1)*(-x)^(k*(k+1)/2) +x*O(x^n)),n)}
%Y Cf. A202144.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Dec 12 2011