login
Coefficients in expansion of E_10*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).
6

%I #24 Mar 11 2023 06:57:07

%S 1,-312,-121680,1004000,37942020,-801594864,6139193600,-11831002560,

%T -151614128250,1346611783000,-4592794000704,3738595861728,

%U 15192491492360,47281379454000,-737660590018560,2662090686805056,-3290770281735027,-4884703150768920

%N Coefficients in expansion of E_10*Delta^2 where Delta is the generating function of Ramanujan's tau function (A000594).

%H Seiichi Manyama, <a href="/A290181/b290181.txt">Table of n, a(n) for n = 2..1000</a>

%H S. C. Milne, <a href="https://arxiv.org/abs/math/0009130">Hankel determinants of Eisenstein series</a>, preprint, arXiv:0009130 [math.NT], 2000.

%F Let b(q) be the determinant of the 3 X 3 matrix [E_6, E_10, E_12 ; E_8, E_12, E_14 ; E_10, E_14, E_16]. G.f. is 691^2*3617*b(q)/(1728^2*2^2*3*5^6*7^2*13).

%t terms = 18;

%t E10[x_] = 1 - 264*Sum[k^9*x^k/(1 - x^k), {k, 1, terms}];

%t E10[x]*QPochhammer[x]^48 + O[x]^terms // CoefficientList[#, x]& (* _Jean-François Alcover_, Feb 26 2018 *)

%Y E_k*Delta^2: A290178 (k=4), A290048 (k=6), A290180 (k=8), this sequence (k=10), A290182 (k=14).

%Y Cf. A000594, A013974 (E_10).

%K sign

%O 2,2

%A _Seiichi Manyama_, Jul 23 2017