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%I M5160 #88 Feb 29 2024 20:33:15
%S 1,24,324,3200,25650,176256,1073720,5930496,30178575,143184000,
%T 639249300,2705114880,10914317934,42189811200,156883829400,
%U 563116739584,1956790259235,6599620022400,21651325216200,69228721526400,216108718571250,659641645039360,1971466420726656
%N Expansion of 1/eta(q)^24; Fourier coefficients of T_{14}.
%C Euler transform of period 1 sequence [24,24,...].
%C Equals A023021 convolved with A000041. - _Gary W. Adamson_, Jun 09 2009
%C Equals convolution square of A005758: (1, 12, 90, 520, 2535, 10908, ...). - _Gary W. Adamson_, Jun 13 2009
%C Note the remarkably wide range of subjects where this sequence appears. - _N. J. A. Sloane_, Oct 29 2019
%D Arnaud Beauville, Counting rational curves on K3 surfaces, arXiv:alg-geom/9701019, Jan 1997.
%D Frenkel, I. B. Representations of Kac-Moody algebras and dual resonance models. Applications of group theory in physics and mathematical physics (Chicago, 1982), 325--353, Lectures in Appl. Math., 21, Amer. Math. Soc., Providence, RI, 1985. MR0789298 (87b:17010).
%D Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table III.
%D C. J. Moreno and A. Rocha-Caridi, The exact formula for the weight multiplicities of affine Lie algebras, I, pp. 111-152 of G. E. Andrews et al., editors, Ramanujan Revisited. Academic Press, NY, 1988.
%D C. L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980, pp. 249-268.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D Vainsencher, Israel. "Enumeration of n-fold tangent hyperplanes to a surface." arXiv preprint alg-geom/9312012 (1993). Section 5.5 appears to give these numbers in the context of enumerating n-nodal curves, a result which was later established by Beauville.
%D S.-T. YAU, E. ZASLOW: BPS states, string duality, and nodal curves on K3. Preprint arXiv:hep-th/9512121, 1995.
%H Seiichi Manyama, <a href="/A006922/b006922.txt">Table of n, a(n) for n = -1..10000</a> (first 202 terms from T. D. Noe)
%H R. E. Borcherds, <a href="http://www.math.berkeley.edu/~reb/papers/">Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras</a>, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.
%H Richard E. Borcherds, <a href="https://amathr.org/Borcherds-vinberg/">Vinberg’s Algorithm and Kac-Moody algebras</a>, Vinberg Lecture, Feb 26 2024.
%H Reinhold W. Gebert, <a href="https://arxiv.org/abs/hep-th/9308151">Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebra</a>, Internat. J. Modern Phys. A 8(1993), no. 31, 5441--5503. MR1248070 (95a:17037) [See Sect. 4.6 - _N. J. A. Sloane_, Apr 07 2014]
%H Vaclav Kotesovec, <a href="/A006922/a006922.jpg">Graph - the asymptotic ratio</a>
%H <a href="/index/Pro#1mxtok">Index entries for expansions of Product_{k >= 1} (1-x^k)^m</a>
%F G.f.: (1/x)(Product_{k>0} (1-x^k))^-24 = 1/Delta (the discriminant in Siegel's notation).
%F a(n) ~ 2*Pi * BesselI(13, 4*Pi*sqrt(n)) / n^(13/2) ~ exp(4*Pi*sqrt(n)) / (sqrt(2)*n^(27/4)) * (1 - 675/(32*Pi*sqrt(n)) + 450225/(2048*Pi^2*n)). - _Vaclav Kotesovec_, Jan 08 2017
%F a(-1) = 1, a(n) = (24/(n+1))*Sum_{k=1..n+1} A000203(k)*a(n-k) for n > -1. - _Seiichi Manyama_, Mar 26 2017
%e T_{14} = 1/q + 24 + 324q + 3200q^2 + 25650q^3 + ....
%p with(numtheory): b:= proc(n) option remember; `if`(n=0, 1, add(add(d*24, d=divisors(j)) *b(n-j), j=1..n)/n) end: a:= n->b(n+1): seq(a(n), n=-1..40); # _Alois P. Heinz_, Oct 17 2008
%t max = 18; f[x_] := (1/x)*Product[1-x^k, {k, 1, max}]^-24; Join[{1}, CoefficientList[ Series[ f[x] - 1/x, {x, 0, max-1}], x]] (* _Jean-François Alcover_, Oct 11 2011 *)
%t CoefficientList[1/QPochhammer[q]^24 + O[q]^40, q] (* _Jean-François Alcover_, Nov 15 2015 *)
%o (PARI) a(n)=if(n<-1,0,n++; polcoeff(eta(x+x*O(x^n))^-24,n))
%o (Julia) # DedekindEta is defined in A000594.
%o A006922List(len) = DedekindEta(len, -24)
%o A006922List(33) |> println # _Peter Luschny_, Mar 10 2018
%Y Cf. A000594, A048057, A048100, A048101, A048110, A048145.
%Y 24th column of A144064. - _Alois P. Heinz_, Oct 17 2008
%Y Cf. A005758, A023021, A000041. - _Gary W. Adamson_, Jun 09 2009
%K nonn,easy,nice
%O -1,2
%A _N. J. A. Sloane_
%E More terms from Barry Brent (barryb(AT)primenet.com)
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