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

Author

Ralf Stephan, May 30 2007

Keywords

• nonn

Comments

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

Examples

			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 + ...
		

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

Crossrefs

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

Programs

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 *)

PARI

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

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).