A129588 Expansion of q^-1 * theta_2(q)^4 in powers of q^2.

16, 64, 96, 128, 208, 192, 224, 384, 288, 320, 512, 384, 496, 640, 480, 512, 768, 768, 608, 896, 672, 704, 1248, 768, 912, 1152, 864, 1152, 1280, 960, 992, 1664, 1344, 1088, 1536, 1152, 1184, 1984, 1536, 1280, 1936, 1344, 1728, 1920, 1440

Offset

0,1

Comments

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

References

K. Bobek, Einleitung in die Theorie der elliptischen Funktionen, Teubner Leipzig, 1884, p. 101.

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

G.f. Sum_{k>=0} a(k)*q^(2*k + 1) = theta2(q)^4 = theta3(q)^4 - theta4(q)^4.

Expansion of 16 * psi(x)^4 in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jun 11 2007

Number of solutions of 2*n + 1 = (x^2 + y^2 + z^2 + w^2) / 4 in odd integers. - Michael Somos, Jun 11 2007

G.f.: 16 * (Product_{k>0} (1 - x^k) * (1 + x^k)^2)^4. - Michael Somos, Jun 11 2007

a(n) = 16*A008438(n) = A000118(n) - A096727(n).

Example

G.f. = 16 + 64*x + 96*x^2 + 128*x^3 + 208*x^4 + 192*x^5 + 224*x^6 + ...
G.f. = 16*q + 64*q^3 + 96*x^5 + 128*q^7 + 208*q^9 + 192*q^11 + 224*q^13 + ...

Mathematica

a[n_]:= SeriesCoefficient[q^(-1/2)*EllipticTheta[2, 0, q^(1/2)]^4, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Apr 15 2018 *)
CoefficientList[Series[x^(-1/2)*EllipticTheta[2, 0, x^(1/2)]^4, {x, 0, 50}], x] (* Vaclav Kotesovec, Apr 16 2018 *)

Prog

(PARI) {a(n) = if( n<0, 0, 16 * sigma(2*n + 1))}; /* Michael Somos, Jun 11 2007 */

Crossrefs

Cf. A000118, A000122, A002448, A008438, A096727.

Sequence in context: A039370 A324880 A304476 * A043193 A043973 A153262

Adjacent sequences: A129585 A129586 A129587 * A129589 A129590 A129591

Keyword

nonn

Author

Ralf Stephan, May 30 2007

Status

approved