A107653 Expansion of q / (chi(q) * chi(q^3))^6 in powers of q where chi() is a Ramanujan theta function.

1, -6, 21, -68, 198, -510, 1248, -2904, 6393, -13604, 28044, -55956, 108982, -207552, 386622, -707216, 1271970, -2250582, 3925780, -6757272, 11483232, -19290824, 32057352, -52722744, 85884503, -138644292, 221885805, -352241792, 554892894

Offset

1,2

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 = 1..1000

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Convolution inverse of A186829. - Michael Somos, Feb 27 2011

Expansion of (eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2) * eta(q^6))^2)^6 in powers of q.

Euler transform of period 12 sequence [-6, 6, -12, 0, -6, 12, -6, 0, -12, 6, -6, 0, ...]. - Michael Somos, Jun 13 2005

G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 27 2011

G.f.: x * (Product_{k>0} ((1 + x^(2*k)) * (1 + x^(6*k))) / ((1 + x^k) * (1 + x^(3*k))))^6 = x * (Product_{k>0} (1 + x^(2*k-1)) * (1 + x^(6*k-3)))^-6.

a(n) = -(-1)^n * A123653(n). - Michael Somos, Feb 27 2011

Example

G.f. = q - 6*q^2 + 21*q^3 - 68*q^4 + 198*q^5 - 510*q^6 + 1248*q^7 + ...

Mathematica

QP = QPochhammer; s = (QP[q]*QP[q^3]*QP[q^4]*(QP[q^12]/(QP[q^2]*QP[q^6])^2 ))^6 + O[q]^30; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
a[n_]:= SeriesCoefficient[q/( QPochhammer[-q, q^2]* QPochhammer[-q^3, q^6])^6, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Dec 09 2017 *)

Prog

(PARI) {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^6 + A)^2))^6, n))}; /* Michael Somos, Jun 13 2005 */

Crossrefs

Cf. A123653, A186829.

Sequence in context: A119103 A180795 A306089 * A123653 A375297 A364636

Adjacent sequences: A107650 A107651 A107652 * A107654 A107655 A107656

Keyword

sign

Author

N. J. A. Sloane, Jun 07 2005

Extensions

Revised by Michael Somos, Jun 12 2005

Status

approved