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

Author

Michael Somos, Dec 24 2012

Keywords

• nonn

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

Examples

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

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.

Crossrefs

Cf. A000041, A076394.

Programs

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 *)

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)}