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

0,2

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.

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

Example

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 + ...

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 *)

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).

Sequence in context: A103924 A160647 A103925 * A103926 A103927 A103928

Adjacent sequences: A160522 A160523 A160524 * A160526 A160527 A160528

Keyword

nonn

Author

N. J. A. Sloane, Nov 13 2009

Status

approved