A122163 Expansion of f(-q)^2*P(q) in powers of q.

1, -26, -25, 74, 49, 122, -146, 0, -194, -218, 121, 0, 0, 314, 507, -362, 386, 0, 0, -458, -482, 0, 0, -554, -289, 0, 626, 650, -674, 698, 361, 746, 0, 794, -818, -842, 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

Offset

0,2

Comments

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

Links

Table of n, a(n) for n=0..61.

Formula

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.

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

Prog

(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))}
(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)))))}

Crossrefs

Sequence in context: A022982 A023468 A261309 * A109341 A010865 A214482

Adjacent sequences: A122160 A122161 A122162 * A122164 A122165 A122166

Keyword

sign

Author

Michael Somos, Aug 22 2006

Status

approved