A160525 Coefficients in the expansion of C/B^2, in Watson's notation of page 118.

1, 2, 5, 10, 20, 36, 65, 109, 183, 295, 471, 732, 1129, 1705, 2554, 3769, 5517, 7979, 11458, 16289, 23007, 32227, 44869, 62028, 85284, 116530, 158432, 214228, 288348, 386224, 515156, 684109, 904963, 1192353, 1565383, 2047642, 2669591, 3468797, 4493351, 5802533

Offset: 0

Author

N. J. A. Sloane, Nov 13 2009

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + 5*x^2 + 10*x^3 + 20*x^4 + 36*x^5 + 65*x^6 + 109*x^7 + ...
G.f. = q^5 + 2*q^29 + 5*q^53 + 10*q^77 + 20*q^101 + 36*q^125 + 65*q^149 + 109*q^173 + ...
		

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.

Crossrefs

Cf. A160526, A160527, A160528.

Cf. Product_{n>=1} (1 - x^(k*n))/(1 - x^n)^2: A000041 (k=1), A015128 (k=2), A278690 (k=3), A160461 (k=5), this sequence (k=7).

Programs

Maple

M1:=1200:
fm:=mul(1-x^n,n=1..M1):
A:=x^(1/7)*subs(x=x^(24/7),fm):
B:=x*subs(x=x^24,fm):
C:=x^7*subs(x=x^168,fm):
t1:=C/B^2;
t2:=series(t1,x,M1);
t3:=subs(x=y^(1/24),t2/x^5);
t4:=series(t3,y,M1/24);
t5:=seriestolist(t4); # A160525

Mathematica

nmax = 50; CoefficientList[Series[Product[(1 - x^(7*k))/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 13 2017 *)

Formula

See Maple code for formula.

G.f.: Product_{n>=1} (1 - x^(7*n))/(1 - x^n)^2. - Seiichi Manyama, Nov 06 2016

a(n) ~ sqrt(13/3) * exp(sqrt(26*n/21)*Pi) / (28*n). - Vaclav Kotesovec, Apr 13 2017