A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.

1, -4, -4, 32, -4, -104, 32, 192, -4, -292, -104, 480, 32, -680, 192, 832, -4, -1160, -292, 1440, -104, -1536, 480, 2112, 32, -2604, -680, 2624, 192, -3368, 832, 3840, -4, -3840, -1160, 4992, -292, -5480, 1440, 5440, -104, -6728, -1536, 7392, 480, -7592, 2112, 8832, 32, -9412, -2604

Offset

0,2

Comments

Number 8 of the 74 eta-quotients listed in Table I of Martin (1996).

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

References

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).

Links

G. C. Greubel, Table of n, a(n) for n = 0..10000

F. Jarvis, H. A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171.

Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.

Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

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

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (u - v)^2 - 4 * u*w * (v - w) * (u - 2*v).

Euler transform of period 4 sequence [ -4, -10, -4, -6, ...].

G.f.: 1 - 4 * Sum_{k>0} A056594(k-1) * k^2 * x^k / (1 - x^k).

Expansion of phi(-q)^2 * phi(-q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007

G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(2*k^2))^4.

G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 128 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A050470.

a(n) = -4 * A002173(n) unless n=0.

Convolution of A000141 and A128692.

Example

G.f. = 1 - 4*q - 4*q^2 + 32*q^3 - 4*q^4 - 104*q^5 + 32*q^6 + 192*q^7 - 4*q^8 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 (QPochhammer[ q] / QPochhammer[ q^4])^2)^2, {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 24 2015 *)

Prog

(PARI) {a(n) = if( n<1, n==0, -4 * sumdiv( n, d, d^2 * kronecker( -4, d)))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^6 / eta(x^4 + A)^4, n))};
(Magma) A := Basis( ModularForms( Gamma1(4), 3), 51); A[1] - 4*A[2]; /* Michael Somos, May 24 2015 */

Crossrefs

Cf. A000141, A002173, A050470, A128692.

Sequence in context: A239601 A196131 A256691 * A138504 A002611 A130188

Adjacent sequences: A120027 A120028 A120029 * A120031 A120032 A120033

Keyword

sign

Author

Michael Somos, Jun 05 2006

Status

approved