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A138810
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Expansion of (8 / 7) * (1 - eta(q)^7 / eta(q^7)) - 7 * (eta(q) * eta(q^7))^3 in powers of q.
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0
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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REFERENCES
| 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)]
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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_{n>0} x^n Sum_{d|n} d^2 kronecker( -7, d). - Michael Somos Feb 24 2011
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EXAMPLE
| q + 5*q^2 - 8*q^3 + 21*q^4 - 24*q^5 - 40*q^6 + q^7 + 85*q^8 + 73*q^9 + ...
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PROG
| (PARI) {a(n) = if( n<1, 0, sumdiv(n, d, d^2 * kronecker(-7, d)))}
(PARI) {a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k, 1], e = A[k, 2]; if( p==7, 1, if(kronecker(p, 7)==1, ((p^2)^(e+1) - 1) / (p^2 - 1), (-(-p^2)^(e+1) + 1) / (p^2 + 1)))))) }
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CROSSREFS
| A138809(n) = -7 * a(n) unless n=0.
Sequence in context: A178675 A034737 A169705 * A105634 A120036 A036381
Adjacent sequences: A138807 A138808 A138809 * A138811 A138812 A138813
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KEYWORD
| sign,mult
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AUTHOR
| Michael Somos, Mar 31 2008
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