A182668 The n-th Fourier coefficient divided by 11 of L_1(tau) defined by A. O. L. Atkin in 1967.

1, 27, 338, 2835, 18566, 101955, 490253, 2121679, 8424520, 31120519, 108082568, 355805844, 1117485594, 3366122862, 9767102571, 27398599802, 74534162438, 197147428426, 508187725366, 1279132093597, 3149343999710, 7596355910693, 17974782074306, 41775768918777

Offset

1,2

Comments

Atkin (1967) on page 22, equation (30), defines phi(tau) = eta(121*tau) / eta(tau), a modular function which satisfies phi(-1/(121*t)) = 11^(-1)/phi(t), where q = exp(2*Pi*i*t). On page 23, equation (33), he defines L_1(tau) = U phi(tau), where U is a Hecke operator so that the n-th Fourier coefficient of L_1 is the 11*n-th Fourier coefficient of phi. On page 26, he finds that L_1(tau) = 11*g_2(tau) + 2*11^2*g_3(tau) + 11^3*g_4(tau) + 11^4*g_5(tau), where g_2, g_3, g_4, g_5 are functions he previously defined. The n-th Fourier coefficient of L_1 is 11*a(n).

First differs from A076394 at a(12). - Omar E. Pol, Dec 24 2012

The sequence of coefficients of the q-expansion of phi(tau) coincides with the partition function A000041 for the first 120 terms. - N. J. A. Sloane, Dec 24 2012

Links

G. C. Greubel, Table of n, a(n) for n = 1..1000

A. O. L. Atkin, Proof of a conjecture of Ramanujan, Glasgow Math. J., 8 (1967), 14-32.

Example

x + 27*x^2 + 338*x^3 + 2835*x^4 + 18566*x^5 + 101955*x^6 + 490253*x^7 + ...

Mathematica

eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[ eta[q^121]/ eta[q]/11, {q, 0, 300}], q][[1 ;; -1 ;; 11]] (* G. C. Greubel, Aug 10 2018 *)

Prog

(PARI) {a(n) = local(A); if( n<1, 0, n = 11*n - 5; A = x * O(x^n); polcoeff( eta(x^121 + A) / eta(x + A), n) / 11)}

Crossrefs

Cf. A000041, A076394.

Sequence in context: A048709 A268973 A160223 * A076394 A133211 A178983

Adjacent sequences: A182665 A182666 A182667 * A182669 A182670 A182671

Keyword

nonn

Author

Michael Somos, Dec 24 2012

Status

approved