A262064 Expansion of f(x^9, x^15) / f(-x^2, -x^4) in powers of x where f(, ) is the Ramanujan general theta function.
1, 0, 1, 0, 2, 0, 3, 0, 5, 1, 7, 1, 11, 2, 15, 4, 22, 6, 30, 9, 42, 14, 56, 20, 77, 29, 101, 41, 135, 57, 176, 78, 231, 107, 297, 143, 385, 191, 490, 253, 627, 332, 793, 432, 1003, 561, 1257, 721, 1578, 924, 1963, 1177, 2443, 1492, 3022, 1882, 3734, 2367, 4589
Offset: 0
Keywords
Examples
G.f. = 1 + x^2 + 2*x^4 + 3*x^6 + 5*x^8 + x^9 + 7*x^10 + x^11 + 11*x^12 + ... G.f. = q^5 + q^101 + 2*q^197 + 3*q^293 + 5*q^389 + q^437 + 7*q^485 + q^533 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..2500
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Crossrefs
Cf. A143067.
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^9, x^24] QPochhammer[ -x^15, x^24] QPochhammer[ x^24] / QPochhammer[ x^2], {x, 0, n}]; -
PARI
{a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( subst( prod(k=1, n\3, 1 - x^k * [1, 0, 0, 1, 0, 1, 0, 0][k%8 + 1], 1 + x * O(x^(n\3))), x, -x^3) / eta(x^2 + x * O(x^n)), n))};
Formula
Euler transform of period 48 sequence [ 0, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, ...].
a(n) = A143067(2*n).
Comments