A256209 Coefficients of mock modular form H_2^(4) (divided by 16).

1, 3, 7, 14, 27, 49, 84, 141, 230, 364, 567, 867, 1302, 1932, 2829, 4091, 5859, 8309, 11675, 16275, 22513, 30914, 42174, 57176, 77049, 103263, 137669, 182616, 241110, 316910, 414750, 540603, 701903, 907928, 1170261, 1503238, 1924607, 2456349

Offset: 0

Author

N. J. A. Sloane, Mar 25 2015

Keywords

• nonn

Comments

The coefficients occur on page 94, Table 24, column 1A for McKay-Thompson series H_{1A,2}^(4) in the Cheng et al. arXiv article. - Michael Somos, Nov 04 2015

Examples

			G.f. = 1 + 3*x + 7*x^2 + 14*x^3 + 27*x^4 + 49*x^5 + 84*x^6 + 141*x^7 + ...
G.f. = q^3 + 3*q^7 + 7*q^11 + 14*q^15 + 27*q^19 + 49*q^23 + 84*q^27 + ...
		

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 2nd equation.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..2500
• Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, Umbral Moonshine, arXiv:1204.2779 [math.RT], 2012-2013.

Crossrefs

Equals A256052/8.

Cf. A053270, A257640.

Programs

Mathematica

nmax = 50; a:= CoefficientList[Series[q*Sum[q^(k - 1)*(Product[1 + q^j, {j, 1, 2 k - 2}])/(Product[1 - q^(2 j - 1), {j, 1, k}])^2, {k, 0, nmax}], {q, 0, 150}], q]; Table[a[[n]], {n, 1, 100}] (* G. C. Greubel, Jul 27 2018 *)

PARI

{a(n) = if( n<0, 0, n++; polcoeff( sum(k=1, n, x^k * prod(i=1, 2*k - 2, 1 + x^i, 1 + x * O(x^(n - k))) / prod(i=1, k, 1 - x^(2*i - 1), 1 + x * O(x^(n - k)))^2), n))}; /* Michael Somos, Nov 04 2015 */

Formula

G.f.: Sum_{k>0} x^(k-1) * (1 + x) * ... * (1 + x^(2*k-2)) / ((1 + x) * (1 + x^3) * ... (1 + x^(2*k-1)))^2. - Michael Somos, Nov 04 2015

2 * a(n) = A053270(3*n) - A257640(3*n). - Michael Somos, Nov 04 2015