A225543 G.f.: Product_{k>0} (1 - x^k)^4 * (1 - (-x)^k)^8.

1, 4, -10, -56, 29, 332, 30, -1064, -302, 1940, 288, -1960, 1071, 1192, -1938, -736, -2000, -1488, 5014, 7288, 4170, -10644, -8482, 11184, -12647, -15544, 15590, 9992, 25424, 4604, -26610, 2472, -28972, 3140, 26464, -39416, 31338, 24764, -25248, -16176

Offset

0,2

Comments

This is Glaisher's alpha(m) for odd values of m. - N. J. A. Sloane, Nov 24 2018

Links

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

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. 37).

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Index entries for sequences mentioned by Glaisher

Formula

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

Expansion of phi(-x^2)^8 * psi(x)^4 in powers of x where phi(), psi() are Ramanujan theta functions. (Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).)

Expansion of f(x)^8 * f(-x)^4 in powers of x where f() is a Ramanujan theta function.

Expansion of q^(-1/2) * (eta(q^2)^6 / (eta(q) * eta(q^4)^2))^4 in powers of q.

Euler transform of period 4 sequence [ 4, -20, 4, -12, ...].

|a(n)| = A002290(n).

Example

1 + 4*x - 10*x^2 - 56*x^3 + 29*x^4 + 332*x^5 + 30*x^6 - 1064*x^7 + ...
or
q + 4*q^3 - 10*q^5 - 56*q^7 + 29*q^9 + 332*q^11 + 30*q^13 - 1064*q^15 + ...

Mathematica

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^4 QPochhammer[ -q]^8, {q, 0, n}]
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^4]^2 EllipticTheta[ 2, 0, q] / 2)^4, {q, 0, 1 + 2 n}]

Prog

(PARI) {a(n) = local(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, n))}

Crossrefs

Cf. A002290.

Sequence in context: A263044 A013589 A367430 * A002290 A336997 A222569

Adjacent sequences: A225540 A225541 A225542 * A225544 A225545 A225546

Keyword

sign

Author

Michael Somos, May 17 2013

Extensions

Entry revised by N. J. A. Sloane, Apr 27 2014

Status

approved