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A160539
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Coefficients in the expansion of C/B^7, in Watson's notation of page 118.
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6
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1, 7, 35, 140, 490, 1547, 4522, 12404, 32298, 80430, 192759, 446656, 1004598, 2199953, 4703104, 9836820, 20167210, 40593651, 80335164, 156503088, 300457906, 568992893, 1063818868, 1965178600, 3589328246, 6485976525, 11602141453, 20555544212, 36087448852
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OFFSET
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0,2
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COMMENTS
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Watson's C and B are (essentially) defined as C = prod(n>=1, 1-q^(7*n)) and B = prod(n>=1, 1-q^n). - Joerg Arndt, Jul 30 2011
In Section 5 of his paper, p. 118, Watson defines A = x^(1/7)*f(-x^(24/7)), B = x*f(-x^24), and C = x^7*f(-x^168), where f(-x^2) = Product_{n >= 1} (1 - x^(2*n)). Note that in different sections of the paper, the definitions of A, B, and C change.
Letting q = x^24, we get B = q^(1/24) * Product_{n >= 1} (1 - q^n), C = q^(7/24) * Product_{n >= 1} (1 - q^(7*n)), and C/B^7 = Product_{n >= 1} (1 - q^(7*n))/(1 -q^n)^7. This is the reason Joerg Arndt above omits the factor q^(1/24) in the definition of B and the factor q^(7/24) in the definition of C.
(End)
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LINKS
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FORMULA
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G.f.: E7/E1^7 where E1 = P(q), E7 = P(q^7), and P(q) = prod(n>=1, 1-q^n). - Joerg Arndt, Jul 30 2011
G.f.: exp(sum(n>=1, (sigma(7*n)-sigma(n))*x^n/n ) ). - Joerg Arndt, Jul 30 2011
See also Maple code in A160525 for formula.
a(n) ~ 2^(5/4) * exp(4*Pi*sqrt(2*n/7)) / (7^(9/4) * n^(9/4)). - Vaclav Kotesovec, Nov 10 2016
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EXAMPLE
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1 + 7*x^24 + 35*x^48 + 140*x^72 + 490*x^96 + 1547*x^120 + 4522*x^144 + ... = 1 + 7*q + 35*q^2 + 140*q^3 + 490*q^4 + 1547*q^5 + ... with q = x^24.
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MATHEMATICA
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nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k)^7, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 10 2016 *)
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PROG
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(PARI) N=66; x='x+O('x^N);
gf=eta(x^7)/eta(x)^7;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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