A259551 Expansion of f(x^2, x^3) * f(-x^4, -x^6) / f(-x^2) in powers of x where f(,) is the Ramanujan general theta function.
1, 0, 2, 1, 2, 1, 2, 1, 3, 2, 4, 4, 5, 4, 6, 5, 7, 6, 9, 8, 11, 11, 13, 13, 17, 15, 20, 19, 23, 23, 27, 27, 33, 33, 38, 39, 45, 45, 53, 54, 62, 63, 73, 74, 84, 86, 97, 100, 112, 115, 130, 134, 148, 154, 170, 176, 195, 202, 222, 232, 255, 264, 290, 301, 329
Offset: 0
Keywords
Examples
G.f. = 1 + 2*x^2 + x^3 + 2*x^4 + x^5 + 2*x^6 + x^7 + 3*x^8 + 2*x^9 + ... G.f. = 1/q + 2*q^239 + q^359 + 2*q^479 + q^599 + 2*q^719 + q^839 + ...
References
- Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 23, 9th equation.
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- G. E. Andrews, An introduction to Ramanujan's "lost" notebook, Amer. Math. Monthly 86 (1979), no. 2, 89-108. See page 97, Equation (3.5)
- Michael Somos, Introduction to Ramanujan theta functions
- Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Programs
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Mathematica
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, -2, -1, 1, 1, 1, -1, -2, 0, 1}[[Mod[k, 10, 1]]], {k, n}], {x, 0, n}]; a[ n_] := SeriesCoefficient[ QPochhammer[x^5] QPochhammer[ -x^2, x^5] QPochhammer[ -x^3, x^5] / (QPochhammer[ x^2, x^10] QPochhammer[ x^8, x^10]), {x, 0, n}]; -
PARI
{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 1, 0, -2, -1, 1, 1, 1, -1, -2, 0][k%10 + 1]), n))};
Formula
Expansion of f(x^2, x^3) * G(x^2) in powers of x where f(,) is the Ramanujan general theta function and G() is a Rogers-Ramanujan function.
Euler transform of period 10 sequence [ 0, 2, 1, -1, -1, -1, 1, 2, 0, -1, ...].
G.f.: Product_{k>0} (1 - x^(5*k)) * (1 + x^(5*k - 3)) * (1 + x^(5*k - 2)) / ((1 - x^(10*k - 8)) * (1 - x^(10*k - 2))).
G.f.: (Sum_{k in Z} x^(k*(5*k + 1)/2)) / (Product_{k in Z} 1 - x^abs(10*k + 2)). - Michael Somos, Jul 09 2015
Comments