A034320 Coefficients of completely replicable function 50a with a(0) = 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

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

G. C. Greubel, Table of n, a(n) for n = -1..1000

D. Alexander, C. Cummins, J. McKay and C. Simons, Completely Replicable Functions, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.

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.

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

H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013; Mentions this sequence.

H. D. Nguyen, D. Taggart, Mining the OEIS: Ten Experimental Conjectures, 2013

Michael 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

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.

a(n) ~ exp(2*Pi*sqrt(2*n)/5) / (2^(3/4) * sqrt(5) * n^(3/4)). - Vaclav Kotesovec, Sep 06 2015

Example

G.f. = 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) = my(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) = my(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 */
(PARI) N=66; q='q+O('q^N); Vec( (eta(q^2)*eta(q^25))/(eta(q)*eta(q^50))/q ) \\ Joerg Arndt, Apr 09 2016

Crossrefs

Cf. A034321, A058703.

Sequence in context: A347587 A288001 A034321 * A058703 A347588 A000009

Adjacent sequences: A034317 A034318 A034319 * A034321 A034322 A034323

Keyword

nonn

Author

N. J. A. Sloane

Status

approved