A058706 McKay-Thompson series of class 52B for Monster.

1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 17, 21, 26, 30, 36, 43, 50, 59, 70, 81, 94, 110, 127, 147, 170, 195, 224, 258, 294, 336, 384, 436, 496, 564, 638, 722, 816, 920, 1037, 1168, 1312, 1473, 1654, 1851, 2072, 2317, 2586, 2886, 3218, 3583, 3986, 4432, 4922, 5462

Offset

0,4

Links

G. C. Greubel, Table of n, a(n) for n = 0..1500

D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Index entries for McKay-Thompson series for Monster simple group

Formula

Expansion of q*eta(q^4)eta(q^26)/(eta(q^2)eta(q^52)) in powers of q^2. - Michael Somos, May 03 2005

Euler transform of period 26 sequence [1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]. - Michael Somos, May 03 2005

Given g.f. A(x), then B(x)=(A(x^2)/x)^2 satisfies 0=f(B(x), B(x^2)) where f(u, v)= u^3 +v^3 -u^3*v^3 -6*u^2*v^2 -u*v +4*u^3*v^2 +4*u^2*v^3 -4*u^3*v -4*u*v^3 +4*u^2*v +4*u*v^2 +u^4*v +u*v^4. - Michael Somos, May 03 2005

Given g.f. A(x), then B(x)=A(x^2)/x satisfies 0=f(B(x), B(x^2), B(x^4)) where f(u, v, w)= v^4 -v^3 -v^2 -u^2*v -w^2*v +u^2*w^2 +u^2*v^2 +w^2*v^2 +u^2*w^2*v +u^2*v^3 +w^2*v^3 -u^2*w^2*v^2. - Michael Somos, May 03 2005

Given g.f. A(x), then B(x)=A(x^2)/x satisfies 0=f(B(x), B(x^2), B(x^3), B(x^6)) where f(u1, u2, u3, u6)=u2^4 -u6^3*u2^3 +3*u6*u2^3 +3*u6^2*u2^2 +3*u6^3*u2 -u6*u2 +u6^4. - Michael Somos, May 03 2005

G.f. Product_{k>0} (1-x^(2k))(1-x^(13k))/((1-x^k)(1-x^(26k))).

a(n) ~ exp(2*Pi*sqrt(n/13)) / (2 * 13^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

Example

T52B = 1/q + q + q^3 + 2*q^5 + 2*q^7 + 3*q^9 + 4*q^11 + 5*q^13 + 6*q^15 + ...

Mathematica

nmax = 50; CoefficientList[Series[Product[(1+x^k) / (1+x^(13*k)), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 06 2015 *)
eta[q_] := q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(1/2)*(eta[q^2]*eta[q^13]/(eta[q]*eta[q^26])), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 27 2018 *)

Prog

(PARI) {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^13+A)/eta(x+A)/eta(x^26+A), n))} /* Michael Somos, May 03 2005 */

Crossrefs

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

Sequence in context: A034142 A008675 A027581 * A034143 A309999 A034144

Adjacent sequences: A058703 A058704 A058705 * A058707 A058708 A058709

Keyword

nonn

Author

N. J. A. Sloane, Nov 27 2000

Status

approved