A138810 Expansion of (8 / 7) * (1 - eta(q)^7 / eta(q^7)) - 7 * (eta(q) * eta(q^7))^3 in powers of q.

1, 5, -8, 21, -24, -40, 1, 85, 73, -120, 122, -168, -168, 5, 192, 341, -288, 365, -360, -504, -8, 610, 530, -680, 601, -840, -656, 21, 842, 960, -960, 1365, -976, -1440, -24, 1533, 1370, -1800, 1344, -2040, -1680, -40, 1850, 2562, -1752, 2650, -2208, -2728, 1, 3005, 2304, -3528, 2810

Offset

1,2

Links

Table of n, a(n) for n=1..53.

A. Alaca, S. Alaca, K. S. Williams, Some Infinite Products of Ramanujan Type, Canad. Math. Bull. Vol. 52 (4), 2009, pp. 481-492, [p. 491, equation (5.1)]

Formula

a(n) is multiplicative and a(7^e) = 1, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1, 2, 4 (mod 7), a(p^e) = (-(-p^2)^(e+1) + 1) / (p^2 + 1) if p == 3, 5, 6 (mod 7).

G.f.: Sum_{k>0} x^k * Sum_{d|k} d^2 * Kronecker(-7, d). - Michael Somos, Feb 24 2011

-7*a(n) = A138809(n) unless n = 0.

Example

G.f. = q + 5*q^2 - 8*q^3 + 21*q^4 - 24*q^5 - 40*q^6 + q^7 + 85*q^8 + 73*q^9 + ...

Prog

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( -7, d)))};
(PARI) {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==7, 1, kronecker( p, 7)==1, ((p^2)^(e+1) - 1) / (p^2 - 1), (-(-p^2)^(e+1) + 1) / (p^2 + 1)))) };

Crossrefs

Cf. A138809.

Sequence in context: A272115 A268195 A169705 * A331700 A105634 A294124

Adjacent sequences: A138807 A138808 A138809 * A138811 A138812 A138813

Keyword

sign,mult

Author

Michael Somos, Mar 31 2008

Status

approved