A261445 Expansion of f(x, x^3) * f(x, x^2)^3 in powers of x where f(, ) is Ramanujan's general theta function.

1, 4, 9, 14, 16, 18, 21, 28, 36, 38, 40, 36, 43, 52, 54, 62, 56, 72, 74, 72, 81, 64, 88, 90, 98, 100, 72, 110, 112, 126, 133, 104, 126, 108, 136, 144, 112, 148, 144, 158, 144, 144, 183, 172, 180, 182, 152, 162, 194, 196, 198, 160, 216, 216, 180, 224, 189, 230

Offset

0,2

Comments

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

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

Expansion of f(-x^2)^3 * phi(-x^3)^3 / phi(-x)^2 in powers of x where phi(), f() are Ramanujan theta functions.

Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^6 / (eta(q)^4 * eta(q^6)^3) in powers of q.

Euler transform of period 6 sequence [4, -1, -2, -1, 4, -4, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260301. - Michael Somos, Nov 13 2015

a(n) = A260109(2*n) = A263021(3*n) = A124815(4*n + 1) = A209613(4*n + 1). - Michael Somos, Nov 13 2015

a(3*n + 1) = 4 * A260165(n). a(3*n + 2) = 9 * A263021(n). - Michael Somos, Nov 13 2015

Example

G.f. = 1 + 4*x + 9*x^2 + 14*x^3 + 16*x^4 + 18*x^5 + 21*x^6 + 28*x^7 + ...
G.f. = q + 4*q^5 + 9*q^9 + 14*q^13 + 16*q^17 + 18*q^21 + 21*q^25 + 28*q^29 + ...

Mathematica

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6]))^3 EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^6 EllipticTheta[ 4, 0, x^3]^3 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 13 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 EllipticTheta[ 4, 0, x^3]^3 / EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* Michael Somos, Nov 13 2015 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^6 / (eta(x + A)^4 * eta(x^6 + A)^3), n))};

Crossrefs

Cf. A124815, A209613, A260109 A260165, A260301, A263021.

Sequence in context: A312987 A050986 A284956 * A034259 A312988 A010452

Adjacent sequences: A261442 A261443 A261444 * A261446 A261447 A261448

Keyword

nonn

Author

Michael Somos, Aug 18 2015

Status

approved