cp's OEIS Frontend

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

%I A138810 #18 Sep 07 2018 10:13:56
%S A138810 1,5,-8,21,-24,-40,1,85,73,-120,122,-168,-168,5,192,341,-288,365,-360,
%T A138810 -504,-8,610,530,-680,601,-840,-656,21,842,960,-960,1365,-976,-1440,
%U A138810 -24,1533,1370,-1800,1344,-2040,-1680,-40,1850,2562,-1752,2650,-2208,-2728,1,3005,2304,-3528,2810
%N A138810 Expansion of (8 / 7) * (1 - eta(q)^7 / eta(q^7)) - 7 * (eta(q) * eta(q^7))^3 in powers of q.
%H A138810 A. Alaca, S. Alaca, K. S. Williams, <a href="http://dx.doi.org/10.4153/CMB-2009-050-5">Some Infinite Products of Ramanujan Type</a>, Canad. Math. Bull. Vol. 52 (4), 2009, pp. 481-492, [p. 491, equation (5.1)]
%F A138810 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).
%F A138810 G.f.: Sum_{k>0} x^k * Sum_{d|k} d^2 * Kronecker(-7, d). - _Michael Somos_, Feb 24 2011
%F A138810 -7*a(n) = A138809(n) unless n = 0.
%e A138810 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 + ...
%o A138810 (PARI) {a(n) = if( n<1, 0, sumdiv( n, d, d^2 * kronecker( -7, d)))};
%o A138810 (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)))) };
%Y A138810 Cf. A138809.
%K A138810 sign,mult
%O A138810 1,2
%A A138810 _Michael Somos_, Mar 31 2008