%I #33 Sep 24 2019 02:34:17

%S 1,7,35,140,490,1547,4522,12404,32298,80430,192759,446656,1004598,

%T 2199953,4703104,9836820,20167210,40593651,80335164,156503088,

%U 300457906,568992893,1063818868,1965178600,3589328246,6485976525,11602141453,20555544212,36087448852

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

%C 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

%C From _Petros Hadjicostas_, Sep 23 2019: (Start)

%C 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.

%C 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.

%C (End)

%H Seiichi Manyama, <a href="/A160539/b160539.txt">Table of n, a(n) for n = 0..1000</a>

%H G. N. Watson, <a href="http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002174499">Ramanujans Vermutung über Zerfällungsanzahlen</a>, 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).

%F 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

%F G.f.: exp(sum(n>=1, (sigma(7*n)-sigma(n))*x^n/n ) ). - _Joerg Arndt_, Jul 30 2011

%F See also Maple code in A160525 for formula.

%F a(n) ~ 2^(5/4) * exp(4*Pi*sqrt(2*n/7)) / (7^(9/4) * n^(9/4)). - _Vaclav Kotesovec_, Nov 10 2016

%e 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.

%t 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 *)

%o (PARI) N=66; x='x+O('x^N);

%o gf=eta(x^7)/eta(x)^7;

%o Vec(gf) /* _Joerg Arndt_, Jul 30 2011 */

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Nov 14 2009