OFFSET
0,3
COMMENTS
Case k=12,i=9 of Gordon Theorem.
REFERENCES
G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, p. 109.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
Expansion of f(-q^9, -q^16) / f(-q) in powers of q where f(,) is the Ramanujan two-variable theta function. - Michael Somos, Apr 29 2008
G.f.: Product_{k>0} (1 - x^(25*k-9)) * (1 - x^(25*k-16)) * (1 - x^(25*k)) / (1 - x^k). - corrected by Vaclav Kotesovec, May 09 2018
a(n) ~ exp(2*Pi*sqrt(11*n/3)/5) * 11^(1/4) * cos(7*Pi/50) / (3^(1/4) * 5^(3/2) * n^(3/4)). - Vaclav Kotesovec, May 09 2018
EXAMPLE
1 + q + 2*q^2 + 3*q^3 + 5*q^4 + 7*q^5 + 11*q^6 + 15*q^7 + 22*q^8 + 29*q^9 + ...
MATHEMATICA
f[x_, y_]:= QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; a:= CoefficientList[Series[f[-q^9, -q^16]/f[-q, -q^2], {q, 0, 100}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Apr 17 2018 *)
nmax = 50; CoefficientList[Series[Product[(1 - x^(25*k - 9))*(1 - x^(25*k - 16))*(1 - x^(25*k))/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, May 09 2018 *)
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( prod(k=1, n\25+1, (1 - x^(25*k-9)) * (1 - x^(25*k-16)) * (1 - x^(25*k)), 1 + x * O(x^n)) / eta(x + x * O(x^n)), n))} /* Michael Somos, Apr 29 2008 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved