A208435 Expansion of x * f(x) * f(-x^12)^3 * psi(x^3) / psi(x^2) in powers of x where psi(), f() are Ramanujan theta functions.
0, 1, 1, -2, 0, 3, -2, -4, 0, 5, 1, -8, 0, 7, -4, -8, 0, 9, 8, -10, 0, 14, -6, -12, 0, 16, 6, -14, 0, 15, -8, -20, 0, 17, 14, -18, 0, 19, -10, -24, 0, 26, 1, -22, 0, 23, -16, -28, 0, 25, 20, -32, 0, 32, -14, -28, 0, 29, 12, -30, 0, 38, -16, -32, 0, 33, 31
Offset: 0
Keywords
Examples
x + x^2 - 2*x^3 + 3*x^5 - 2*x^6 - 4*x^7 + 5*x^9 + x^10 - 8*x^11 + ... q^5 + q^8 - 2*q^11 + 3*q^17 - 2*q^20 - 4*q^23 + 5*q^29 + q^32 - 8*q^35 + ...
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
Crossrefs
Cf. A207541.
Programs
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Mathematica
a[n_]:=SeriesCoefficient[((QP[q^2]^4*QP[q^6]^2*QP[q^12]^3)/(QP[q]*QP[q^3]* QP[q^4]^3)), {q, 0, n}]; Table[a[n], {n, -1, 50}] (* G. C. Greubel, Dec 17 2017 *) -
PARI
{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^6 + A)^2 * eta(x^12 + A)^3 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^3), n))}
Formula
Expansion of q^(-2/3) * eta(q^2)^4 * eta(q^6)^2 * eta(q^12)^3 / (eta(q) * eta(q^3) * eta(q^4)^3) in powers of q.
Euler transform of period 12 sequence [ 1, -3, 2, 0, 1, -4, 1, 0, 2, -3, 1, -4, ...].
a(4*n) = 0. 24 * a(n) = A207541(3*n + 2).
Comments