OFFSET
0,4
COMMENTS
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 7, Equ. (1.4). MR0858826 (88b:11063).
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
FORMULA
Expansion of f(-x^2, -x^8) / f(-x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.
Euler transform of period 10 sequence [ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, ...].
G.f.: ( Sum_{k>=0} x^(2*(k^2+k)) / ((1 - x^2) * (1 - x^4) * ... * (1 - x^(2*k))) ) / Product_{k>0} 1 - x^(2*k-1).
G.f.: Sum_{k>=0} x^(k*(3*k+3)/2) * (1 + x) * (1 + x^2) * ... * (1 + x^(2*k)) / ( (1 - x) * (1 - x^2) * ... * (1 - x^(2*k+1)) ).
EXAMPLE
G.f. = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + 6*x^6 + 8*x^7 + 10*x^8 + 14*x^9 + ...
G.f. = q^49 + q^169 + q^289 + 2*q^409 + 3*q^529 + 4*q^649 + 6*q^769 + 8*q^889 + ...
MATHEMATICA
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^10] QPochhammer[ x^8, x^10] QPochhammer[ x^10] / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Oct 26 2015 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 1, 0, 1, 1, 1, 1, 1, 0, 1, 0}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, Oct 26 2015 *)
PROG
(PARI) {a(n) = my(t); if( n<0, 0, t = 1 / (1 - x) + x * O(x^n); polcoeff( sum(k=1, (sqrtint(24*n + 1) - 1)\6, t = t * x^(3*k) / ((1 - x^k) * (1 - x^(2*k + 1))) + x * O(x^n), t), n))};
(PARI) {a(n) = if( n<0, 0, polcoeff( 1 / prod( k=1, n, 1 - [ 0, 1, 0, 1, 1, 1, 1, 1, 0, 1][k%10 + 1] * x^k, 1 + x * O(x^n)), n))};
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael Somos, Oct 10 2007
STATUS
approved