|
| |
|
|
A034320
|
|
McKay-Thompson series of class 50a for the Monster group with a(0) = 1.
|
|
1
| |
|
|
1, 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, 27, 32, 38, 46, 54, 64, 76, 89, 104, 122, 141, 164, 191, 220, 254, 293, 336, 385, 442, 504, 575, 656, 745, 846, 960, 1086, 1228, 1388, 1564, 1762, 1984, 2228, 2501, 2806, 3142, 3516, 3932, 4390, 4898, 5462, 6082
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| -1,4
|
|
|
COMMENTS
| Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
|
|
|
REFERENCES
| F. Calegari, Review of "A first Course in modular forms" by F. Diamond and J. Shurman, Bull. Amer. Math. Soc., 43 (No. 3, 2006), 415-421. See p. 418
|
|
|
LINKS
| M. Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for McKay-Thompson series for Monster simple group
I. Chen and N. Yui, Singular values of Thompson series. In Groups, difference sets and the Monster (Columbus, OH, 1993), pp. 255-326, Ohio State University Mathematics Research Institute Publications, 4, de Gruyter, Berlin, 1996.
|
|
|
FORMULA
| Expansion of Hauptmodul for Gamma_0(50)+50 in powers of q.Expansion of (q^-1) * chi(-q^25) / chi(-q) in powers of q where chi() is a Ramanujan theta function. - Michael Somos Jun 09 2007
Expansion of (eta(q^2) * eta(q^25)) / (eta(q) * eta(q^50)) in powers of q. - Michael Somos, Sep 20 2004
Euler transform of period 50 sequence [ 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 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, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, ...]. - Michael Somos, Sep 20 2004
G.f. is Fourier series of a weight 0 level 50 modular form. f(-1 / (50 t)) = f(t) where q = exp(2 pi i t). - Michael Somos Jun 09 2007
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2*v + 2*u*w + 2*u*v^2*w + v*w^2 - v^2 - u^2*w^2. - Michael Somos Jun 09 2007
G.f.: 1/x * (Product_{k>0} (1 + x^k) / (1 + x^(25*k))).
a(n) = A058703(n) unless n=0.
|
|
|
EXAMPLE
| q^-1 + 1 + q + 2*q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 5*q^6 + 6*q^7 + 8*q^8 + ...
|
|
|
MATHEMATICA
| a[ n_] := SeriesCoefficient[ q^-1 QPochhammer[q^25, q^50] / QPochhammer[q, q^2], {q, 0, n}] (* Michael Somos Jul 11 2011 *)
a[ n_] := SeriesCoefficient[ q^-1 Product[1 + q^k, {k, n + 1}] / Product[1 + q^k, {k, 25, n + 1, 25}], {q, 0, n}] (* Michael Somos Jul 11 2011 *)
|
|
|
PROG
| (PARI) {a(n) = local(A); if( n<-1, 0, n++; A = 1 + x * O(x^n); polcoeff( prod( k=1, n, 1 + x^k, A) / prod( k=1, n\25, 1 + x^(25*k), A), n))} /* Michael Somos, Sep 20 2004 */
(PARI) {a(n) = local(A); if( n<-1, 0, n++; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^25 + A) / (eta(x + A) * eta(x^50 + A)), n))} /* Michael Somos, Sep 20 2004 */
|
|
|
CROSSREFS
| Cf. A034321, A058703.
Sequence in context: A034150 A034321 A058703 * A000009 A081360 A117409
Adjacent sequences: A034317 A034318 A034319 * A034321 A034322 A034323
|
|
|
KEYWORD
| nonn
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
| |
|
|
