A124863 Expansion of 1 / chi(q)^12 in powers of q where chi() is a Ramanujan theta function.

1, -12, 78, -376, 1509, -5316, 16966, -50088, 138738, -364284, 913824, -2203368, 5130999, -11585208, 25444278, -54504160, 114133296, -234091152, 471062830, -931388232, 1811754522, -3471186596, 6556994502, -12222818640, 22502406793

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 q^(-1/2) * (k * k') / 4 in powers of q where k is the Jacobian elliptic modulus, k' the complementary modulus and q is the nome.

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

Euler transform of period 4 sequence [ -12, 12, -12, 0, ...].

G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Jul 22 2011

G.f.: Product_{k>0} (1 + (-x)^k)^12 = Product_{k>0} 1/(1 + x^(2*k - 1))^12. [corrected by Vaclav Kotesovec, Nov 16 2017]

a(n) = (-1)^n * A022577(n). Convolution inverse of A112142. Convolution square is A100130.

G.f.: T(0), where T(k) = 1 - 1/(1 - 1/(1 - 1/(1+(-x)^(k+1))^12/T(k+1) )); (continued fraction). - Sergei N. Gladkovskii, Nov 06 2013

a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)) / (128*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

G.f.: exp(-12*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018

Example

G.f. = 1 - 12*x + 78*x^2 - 376*x^3 + 1509*x^4 - 5316*x^5 + 16966*x^6 - 50088*x^7 + ...
G.f. = q - 12*q^3 + 78*q^5 - 376*q^7 + 1509*q^9 - 5316*q^11 + 16966*q^13 - 50088*q^15 + ...

Mathematica

a[ n_] := With[ {m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ ((1 - m) m/16/q)^(-1/2), {q, 0, n}]]; (* Michael Somos, Jul 22 2011 *)
a[ n_] := SeriesCoefficient[1/Product[1 + q^k, {k, 1, n, 2}]^12, {q, 0, n}]; (* Michael Somos, Jul 22 2011, fixed by Vaclav Kotesovec, Nov 16 2017 *)
nmax = 30; CoefficientList[Series[Product[1/(1 + x^(2*k - 1))^12, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A) / eta(x^2 + A)^2)^12, n))};

Crossrefs

Cf. A022577, A100130, A112142.

Sequence in context: A290894 A121665 A341225 * A022577 A321949 A227707

Adjacent sequences: A124860 A124861 A124862 * A124864 A124865 A124866

Keyword

sign

Author

Michael Somos, Nov 10 2006

Status

approved