A122163 - OEIS

%I #2 Apr 30 2014 01:37:09

%S 1,-26,-25,74,49,122,-146,0,-194,-218,121,0,0,314,507,-362,386,0,0,

%T -458,-482,0,0,-554,-289,0,626,650,-674,698,361,746,0,794,-818,-842,

%U 866,0,-914,0,-1924,0,0,0,529,-1082,0,0,1154,0,1202,1226,625,-1274,0,1322,1346,0,0,-1418,0,-1466

%N Expansion of f(-q)^2*P(q) in powers of q.

%C f(-q) (g.f. A010815) and P(q) (g.f. A006352) are Ramanujan q-series.

%F a(n)=b(12n+1) where b(n) is multiplicative and b(2^e) = b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2*p^e if p == 7,11 (mod 12), b(p^e) = (-1)^(e/2)(1+(-1)^e)/2*p^e if p == 5 (mod 12), b(p^e) = (e+1)*((-1)^y*p)^e where p == 1 (mod 12) and p = x^2+9y^2.

%F G.f.: (1 -24*Sum_{k>0} x^k/(1-x^k)^2)(Product_{k>0} 1-x^k)^2.

%o (PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*sum(k=1, n, -24*sigma(k)*x^k, 1+A), n))}

%o (PARI) {a(n)= local(A, p, e, y); if(n<0, 0, n=12*n+1; A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p<5, 0, if(p%12>1, if(e%2, 0, ((-1)^(p%12==5)*p^2)^(e/2)), for(i=1, sqrtint(p\9), if(issquare(p-9*i^2), y=i; break)); (e+1)*((-1)^y*p)^e)))))}

%K sign

%O 0,2

%A _Michael Somos_, Aug 22 2006