OFFSET
0,9
COMMENTS
In Ramanujan's lost notebook the generating function is denoted by psi(q) on pages 18 and 20, however on page 20 there is a "-1" first term.
REFERENCES
Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
FORMULA
G.f.: Sum_{k>=0} x^(5*k^2) / ((1 - x^2) * (1 - x^3) * (1 - x^7) * (1 - x^8)...(1 - x^(5*k+2))).
G.f.: ( Sum_{k in Z} -(-1)^k * x^(5*k*(3*k + 1)/2 - 2) / (1 - x^(5*k - 2)) ) / ( Sum_{k in Z} (-1)^k * x^(5*k*(3*k - 1)) ).
a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(3/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019
EXAMPLE
G.f. = 1 + x^2 + x^4 + x^5 + x^6 + x^7 + 2*x^8 + x^9 + 2*x^10 + 2*x^11 + ...
MATHEMATICA
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ x^(5 k^2) / (QPochhammer[ x^2, x^5, k + 1] QPochhammer[ x^3, x^5, k]) // FunctionExpand, {k, 0, Sqrt[n/5]}], {x, 0, n}]];
a[ n_] := If[ n < 0, 0, With[ {m = Sqrt[1 + 24(n + 2)/5]}, SeriesCoefficient[ Sum[ -(-1)^k x^(5 k (3 k + 1)/2 - 2) / (1 - x^(5 k - 2)), {k, Quotient[m + 1, -6], Quotient[m - 1, 6]}] / QPochhammer[ x^5], {x, 0, n}]]];
PROG
(PARI) {a(n) = if( n<0, 0, polcoeff( sum(k=0, sqrtint(n\5), x^(5*k^2) / prod(i=1, 5*k+2, 1 - if( i%5==2 || i%5==3, x^i), 1 + x * O(x^(n - 5*k^2)))), n))};
(PARI) {a(n) = my(A, m); if( n<0, 0, m = sqrtint(1 + 24*(n+2)\5); A = x * O(x^n); polcoeff( sum(k=(m + 1)\-6, (m - 1)\6, -(-1)^k * x^(5*k*(3*k + 1)/2 - 2) / (1 - x^(5*k - 2)), A) / eta(x^5 + A), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Feb 18 2017
STATUS
approved