A058565 - OEIS

%I #30 Mar 12 2021 22:24:42

%S 1,3,8,11,25,35,57,86,139,198,291,417,588,812,1132,1538,2103,2805,

%T 3767,4963,6554,8548,11165,14426,18601,23830,30443,38642,48986,61748,

%U 77669,97206,121478,151067,187556,231974,286385,352340,432641,529688,647241,788738,959470,1164291,1410386

%N McKay-Thompson series of class 21C for the Monster group.

%D B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 176 Entry 32(iii).

%H G. C. Greubel, <a href="/A058565/b058565.txt">Table of n, a(n) for n = 0..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>, Comm. Algebra 22, No. 13, 5175-5193 (1994).

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

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

%F From _Michael Somos_, Feb 26 2017: (Start)

%F Expansion of f(-x^7, -x^14)^2 / f(-x, -x^2) * (w3/w1^2 + x*w2/w3^2 - x*w1/w2^2) in powers of x where w1 = f(-x, -x^6), w2 = f(-x^2, -x^5), w3 = f(-x^3, -x^4) and f(, ) is Ramanujan's general theta function.

%F G.f. is a period 1 Fourier series which satisfies f(-1 / (63 t)) = f(t) where q = exp(2 Pi i t).

%F Convolution cube is A282877.

%F Convolution product with A002655 is A002652. (End)

%F Expansion of A + 4*q/A^2, where A = q^(1/3)*(eta(q)*eta(q^7)/(eta(q^2) *eta(q^14))), in powers of q. - _G. C. Greubel_, Jun 21 2018

%F a(n) ~ exp(4*Pi*sqrt(n/21)) / (sqrt(2) * 21^(1/4) * n^(3/4)). - _Vaclav Kotesovec_, Feb 26 2017

%e G.f. = 1 + 3*x + 8*x^2 + 11*x^3 + 25*x^4 + 35*x^5 + 57*x^6 + 86*x^7 + ... -  _Michael Somos_, Feb 26 2017

%e T21C = 1/q + 3*q^2 + 8*q^5 + 11*q^8 + 25*q^11 + 35*q^14 + 57*q^17 + ...

%t a[ n_] := With[ {A = (QPochhammer[ x^7] / QPochhammer[ x])^4}, SeriesCoefficient[ (1/A + 13 x + 49 x^2 A)^(1/3), {x, 0, n}]]; (*  _Michael Somos_, Feb 26 2017 *)

%t eta[q_]:= q^(1/24)*QPochhammer[q]; A:= q^(1/3)*(eta[q]*eta[q^7]/(eta[q^2] *eta[q^14])); a:= CoefficientList[Series[(A + 4*q/A^2), {q,0,60}], q]; Table[a[[n]], {n, 1, 50}] (* _G. C. Greubel_, Jun 21 2018 *)

%t a[ n_] := With[ {A1 = QPochhammer[ x] QPochhammer[ x^7], A2 = QPochhammer[ x^2] QPochhammer[ x^14]}, SeriesCoefficient[ (A1^3 + 4 x A2^3) / (A1^2 A2), {x, 0, n}]]; (*  _Michael Somos_, Oct 27 2018 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); A = (eta(x^7 + A) / eta(x + A))^4; polcoeff( (1/A + 13*x + 49*x^2 * A)^(1/3), n))}; /*  _Michael Somos_, Feb 26 2017 */

%o (PARI) q='q+O('q^50); A = (eta(q)*eta(q^7)/(eta(q^2) *eta(q^14))); Vec(A + 4*q/A^2) \\ _G. C. Greubel_, Jun 21 2018

%o (PARI) {a(n) = my(A, A1, A2); if( n<0, 0, A = x * O(x^n); A1 = eta(x + A) * eta(x^7 + A); A2 = eta(x^2 + A) * eta(x^14 + A); polcoeff( (A1^3 + 4 * x * A2^3) / (A1^2 * A2), n))}; /* _Michael Somos_, Oct 27 2018 */

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

%Y Cf. A002652, A002655, A282877.

%K nonn

%O 0,2

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

%E Terms a(8) onward added by _G. C. Greubel_, Jun 21 2018