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McKay-Thompson series of class 24F for Monster.
3

%I #24 Jun 29 2018 09:41:53

%S 1,3,6,10,15,24,37,57,84,118,165,228,316,432,582,776,1023,1344,1757,

%T 2283,2946,3774,4812,6108,7725,9732,12204,15240,18957,23508,29065,

%U 35826,44022,53924,65868,80256,97557,118305,143118,172726,208002,249972,299825,358926,428844,511416,608796

%N McKay-Thompson series of class 24F for Monster.

%H G. C. Greubel, <a href="/A058576/b058576.txt">Table of n, a(n) for n = -1..1000</a>

%H D. Ford, J. McKay and S. P. Norton, <a href="http://dx.doi.org/10.1080/00927879408825127">More on replicable functions</a>, Commun. Algebra 22, No. 13, 5175-5193 (1994).

%H <a href="/index/Mat#McKay_Thompson">Index entries for McKay-Thompson series for Monster simple group</a>

%F a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(5/4) * 3^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Jun 28 2018

%e G.f. = 1 + 3*x + 6*x^2 + 10*x^3 + 15*x^4 + 24*x^5 + 37*x^6 + 57*x^7 + 84*x^8 + ...

%e T24F = 1/q + 3*q^3 + 6*q^7 + 10*q^11 + 15*q^15 + 24*q^19 + 37*q^23 + 57*x^27 + ...

%t eta[q_] := q^(1/24)*QPochhammer[q]; e24F := q^(1/4)*(eta[q^2]*eta[q^3]/(eta[q]*eta[q^6]))^3; Table[SeriesCoefficient[e24F, {q,0,n}], {n, 0, 50}] (* _G. C. Greubel_, Feb 14 2018 *)

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^3] / (QPochhammer[ x] QPochhammer[ x^6]))^3, {x, 0, n}]; (* _Michael Somos_, Feb 18 2018 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) / (eta(x + A) * eta(x^6 + A)))^3, n))}; /* _Michael Somos_, Feb 18 2018 */

%Y Cf. A000521, A007240, A014708, A007241, A007267, A045478, etc.

%K nonn

%O -1,2

%A _N. J. A. Sloane_, Nov 27 2000

%E Terms a(6) onward added by _G. C. Greubel_, Feb 14 2018