A034320 - OEIS

%I #55 Aug 06 2024 04:40:31

%S 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,

%T 141,164,191,220,254,293,336,385,442,504,575,656,745,846,960,1086,

%U 1228,1388,1564,1762,1984,2228,2501,2806,3142,3516,3932,4390,4898,5462,6082

%N Coefficients of completely replicable function 50a with a(0) = 1.

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D 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

%H G. C. Greubel, <a href="/A034320/b034320.txt">Table of n, a(n) for n = -1..1000</a>

%H D. Alexander, C. Cummins, J. McKay and C. Simons, <a href="http://oeis.org/A007242/a007242_1.pdf">Completely Replicable Functions</a>, LMS Lecture Notes, 165, ed. Liebeck and Saxl (1992), 87-98, annotated and scanned copy.

%H I. Chen and N. Yui, <a href="http://www.math.sfu.ca/~ichen/pub.html">Singular values of Thompson series</a>. 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.

%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 H. D. Nguyen, D. Taggart, <a href="https://citeseerx.ist.psu.edu/pdf/8f2f36f22878c984775ed04368b8893879b99458">Mining the OEIS: Ten Experimental Conjectures</a>, 2013; Mentions this sequence.

%H H. D. Nguyen, D. Taggart, <a href="http://users.rowan.edu/~nguyen/papers/Mining%20the%20Online%20Encyclopedia%20of%20Integer%20Sequences.pdf">Mining the OEIS: Ten Experimental Conjectures</a>, 2013

%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 Expansion of Hauptmodul for Gamma_0(50)+50 in powers of q.

%F 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

%F Expansion of (eta(q^2) * eta(q^25)) / (eta(q) * eta(q^50)) in powers of q. - _Michael Somos_, Sep 20 2004

%F 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

%F 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

%F 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

%F G.f.: 1/x * (Product_{k>0} (1 + x^k) / (1 + x^(25*k))).

%F a(n) = A058703(n) unless n=0.

%F a(n) ~ exp(2*Pi*sqrt(2*n)/5) / (2^(3/4) * sqrt(5) * n^(3/4)). - _Vaclav Kotesovec_, Sep 06 2015

%e 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 + ...

%t a[ n_] := SeriesCoefficient[ q^-1 QPochhammer[q^25, q^50] / QPochhammer[q, q^2], {q, 0, n}]; (* _Michael Somos_, Jul 11 2011 *)

%t 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 *)

%o (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 */

%o (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 */

%o (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

%Y Cf. A034321, A058703.

%K nonn

%O -1,4

%A _N. J. A. Sloane_