A225923 Expansion of q^(-1/2) * k(q) * (1 - k(q)^4) * (K(q) / (Pi/2))^6 / 4 in powers of q where k(), k'(), K() are Jacobi elliptic functions.

1, 20, -74, -24, 157, 124, 478, -1480, -1198, 3044, -480, 184, 2351, -1720, -3282, -5728, 2480, 1776, 10326, 9560, -8886, -9188, -11618, 23664, -16231, -23960, 11686, -9176, 60880, 16876, -18482, -3768, -35372, -15532, 3680, -31960, -4886, 47020, -2976, 44560

Offset

0,2

Comments

In Glaisher (1907) denoted by gamma(m) defined in section 63 on page 38.

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

In Chan and Combes (2024) page 2 is g(z) = eta(z)^8 eta(4z)^4 + 8 eta(4z)^12 identified as the unique newform of weight 6 and level 8 with LMFDB label 8.6.a.a. - Michael Somos, Jun 25 2025

References

J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62 (see p. 38).

Links

G. C. Greubel, Table of n, a(n) for n = 0..1000

Elisabeth (Yin Ting) Chan and Lewis Combes, Expressions for Weight 2 Cusp Forms in Holomorphic Eta Quotients, arXiv preprint arXiv:2407.05748 [math.NT], 2024.

LMFDB, Newform 8.6.a.a.

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of (psi(x) * phi(-x^2)^2)^4 + 16 * x * (psi(x) * psi(-x)^2)^4 in powers of x where phi(), psi() are Ramanujan theta functions.

Expansion of (phi(x)^8 - 256 * x^2 * psi(x^2)^8) * psi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jul 20 2013

Expansion of q^(-1/2) * (eta(q)^12 + 32 * q * eta(q)^4 * eta(q^4)^8) in powers of q.

Expansion of q^(-1) * eta(q^4)^4 * (eta(q)^8 + 8 * eta(q^4)^8) in power of q^2. - Michael Somos, Jun 25 2025

G.f. is a period 1 Fourier series which satisfies f(-1/(8*t)) = 512 * (t/i)^6 * f(t) where q = exp(2*Pi*i*t).

a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^5 * b(p^(e-2)) if p > 2.

G.f.: Product_{k>0} (1 - x^k)^12 + 32 * x * (Product_{k>0} (1 - x^k) * (1 - x^(4*k))^2)^4.

|a(n)| = A002292(n). a(n) = A000735(n) + 32 * A225872(n).

Empirical: Sum_{n>=0} a(n) / exp(n*Pi) = (3/32) * exp(Pi / 2) * Pi^3 * sqrt(2) / Gamma(3/4)^12 = A388785. - Simon Plouffe, Sep 18 2025

Example

G.f. = 1 + 20*x - 74*x^2 - 24*x^3 + 157*x^4 + 124*x^5 + 478*x^6 - 1480*x^7 + ...
G.f. = q + 20*q^3 - 74*q^5 - 24*q^7 + 157*q^9 + 124*q^11 + 478*q^13 - 1480*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^12 + 32 q (QPochhammer[ q] QPochhammer[ q^4]^2)^4, {q, 0, n}];
a[ n_] := SeriesCoefficient[ 8 q QPochhammer[ q^4]^12 + (QPochhammer[ q]^2 QPochhammer[ q^4])^4, {q, 0, 2 n}];

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^12 + 32 * x * eta(x + A)^4 * eta(x^4 + A)^8, n))};
(PARI) {a(n) = my(A); if( n<0, 0, n*=2; A = x * O(x^n); polcoeff( 8 * x * eta(x^4 + A)^12 + eta(x + A)^8 * eta(x^4 + A)^4, n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^6 / (eta(x + A) * eta(x^4 + A)^2))^4 + 16 * x * (eta(x + A) * eta(x^4 + A)^2)^4, n))};
(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2  + A)^12 / (eta(x + A)^5 * eta(x^4 + A)^4))^4 - x^2 * (4 * eta(x^4 + A)^4 / eta(x + A))^4, n))}; /* Michael Somos, Jul 20 2013 */

Crossrefs

Cf. A000735, A002292, A225872.

Sequence in context: A139241 A139232 A002292 * A238026 A010008 A237617

Adjacent sequences: A225920 A225921 A225922 * A225924 A225925 A225926

Keyword

sign

Author

Michael Somos, May 20 2013

Status

approved