A261445 Expansion of f(x, x^3) * f(x, x^2)^3 in powers of x where f(, ) is Ramanujan's general theta function.
1, 4, 9, 14, 16, 18, 21, 28, 36, 38, 40, 36, 43, 52, 54, 62, 56, 72, 74, 72, 81, 64, 88, 90, 98, 100, 72, 110, 112, 126, 133, 104, 126, 108, 136, 144, 112, 148, 144, 158, 144, 144, 183, 172, 180, 182, 152, 162, 194, 196, 198, 160, 216, 216, 180, 224, 189, 230
Offset: 0
Keywords
Examples
G.f. = 1 + 4*x + 9*x^2 + 14*x^3 + 16*x^4 + 18*x^5 + 21*x^6 + 28*x^7 + ... G.f. = q + 4*q^5 + 9*q^9 + 14*q^13 + 16*q^17 + 18*q^21 + 21*q^25 + 28*q^29 + ...
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- 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[ (QPochhammer[ x^3] / (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6]))^3 EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^6 EllipticTheta[ 4, 0, x^3]^3 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 13 2015 *) a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 EllipticTheta[ 4, 0, x^3]^3 / EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* Michael Somos, Nov 13 2015 *) -
PARI
{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^6 / (eta(x + A)^4 * eta(x^6 + A)^3), n))};
Formula
Expansion of f(-x^2)^3 * phi(-x^3)^3 / phi(-x)^2 in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^6 / (eta(q)^4 * eta(q^6)^3) in powers of q.
Euler transform of period 6 sequence [4, -1, -2, -1, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260301. - Michael Somos, Nov 13 2015
a(n) = A260109(2*n) = A263021(3*n) = A124815(4*n + 1) = A209613(4*n + 1). - Michael Somos, Nov 13 2015
Comments