A160539 Coefficients in the expansion of C/B^7, in Watson's notation of page 118.

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

Offset: 0

Author

N. J. A. Sloane, Nov 14 2009

Keywords

• nonn

Comments

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

From Petros Hadjicostas, Sep 23 2019: (Start)

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)

Examples

			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.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..1000
• G. N. Watson, Ramanujans Vermutung über Zerfällungsanzahlen, J. Reine Angew. Math. (Crelle), 179 (1938), 97-128; see p. 118 (def. of A, B, C) and p. 125 (expansion of C/B^7).

Programs

Mathematica

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 *)

PARI

N=66; x='x+O('x^N);
gf=eta(x^7)/eta(x)^7;
Vec(gf) /* Joerg Arndt, Jul 30 2011 */

Formula

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