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

%I A256209 #26 Feb 20 2019 09:44:54
%S A256209 1,3,7,14,27,49,84,141,230,364,567,867,1302,1932,2829,4091,5859,8309,
%T A256209 11675,16275,22513,30914,42174,57176,77049,103263,137669,182616,
%U A256209 241110,316910,414750,540603,701903,907928,1170261,1503238,1924607,2456349
%N A256209 Coefficients of mock modular form H_2^(4) (divided by 16).
%C A256209 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
%D A256209 Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 3, 2nd equation.
%H A256209 G. C. Greubel, <a href="/A256209/b256209.txt">Table of n, a(n) for n = 0..2500</a>
%H A256209 Miranda C. N. Cheng, John F. R. Duncan, Jeffrey A. Harvey, <a href="https://arxiv.org/abs/1204.2779">Umbral Moonshine</a>, arXiv:1204.2779 [math.RT], 2012-2013.
%F A256209 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
%F A256209 2 * a(n) = A053270(3*n) - A257640(3*n). - _Michael Somos_, Nov 04 2015
%e A256209 G.f. = 1 + 3*x + 7*x^2 + 14*x^3 + 27*x^4 + 49*x^5 + 84*x^6 + 141*x^7 + ...
%e A256209 G.f. = q^3 + 3*q^7 + 7*q^11 + 14*q^15 + 27*q^19 + 49*q^23 + 84*q^27 + ...
%t A256209 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 *)
%o A256209 (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 */
%Y A256209 Equals A256052/8.
%Y A256209 Cf. A053270, A257640.
%K A256209 nonn
%O A256209 0,2
%A A256209 _N. J. A. Sloane_, Mar 25 2015