A213791 - OEIS

A213791

Expansion of psi(-x)^6 in powers of x where psi() is a Ramanujan theta function.

1, -6, 15, -26, 45, -66, 82, -120, 156, -170, 231, -276, 290, -390, 435, -438, 561, -630, 651, -780, 861, -842, 1020, -1170, 1095, -1326, 1431, -1370, 1716, -1740, 1682, -2016, 2145, -2132, 2415, -2550, 2353, -2850, 3120, -2810, 3321, -3486, 3285, -3906, 4005

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OFFSET

0,2

COMMENTS

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

REFERENCES

J. W. L. Glaisher, Identities, Messenger of Mathematics, 5 (1876), pp. 111-112. see Eq. X

J. W. L. Glaisher, Notes on Certain Formulae in Jacobi's Fundamenta Nova, Messenger of Mathematics, 5 (1876), pp. 174-179. see p.176

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

FORMULA

Expansion of q^(-3/4) * ( eta(q) * eta(q^4) / eta(q^2) )^6 in powers of q.

Expansion of -1/(8 * r) * ( 1^2 * r^1 / (1 + q) - 3^2 * q^(3/4) / (1 + q^3) - 5^2 * r^5 / (1 + q^5) + 7^2 * q^(7/4) / (1 + q^7) + 9^2 * r^9 / (1 + q^9) - ...) in powers of q where r = q^(3/4) [Glaisher 1876].

Expansion of q^(-1/4) * ( sqrt(k * k') * K / Pi )^3 in powers of q where k, k', K are Jacobi elliptic functions. [Jacobi 1828, p. 108 quoted in Glaisher 1876, p. 176].

Euler transform of period 4 sequence [ -6, 0, -6, -6, ...].

G.f.: (Sum_{k>0} (-x)^((k^2 - k)/2))^6.

G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t).

a(n) = (-1)^n * A008440(n). Convolution cube of A134343.

EXAMPLE

G.f. = 1 - 6*x + 15*x^2 - 26*x^3 + 45*x^4 - 66*x^5 + 82*x^6 - 120*x^7 + ...

G.f. = q^3 - 6*q^7 + 15*q^11 - 26*q^15 + 45*q^19 - 66*q^23 + 82*q^27 + ...

MATHEMATICA

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] / QPochhammer[ x^2, x^4])^6, {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)

PROG

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( ( eta(x + A) * eta(x^4 + A) / eta(x^2 + A) )^6, n))};

CROSSREFS