A006922 Expansion of 1/eta(q)^24; Fourier coefficients of T_{14}. (Formerly M5160)

1, 24, 324, 3200, 25650, 176256, 1073720, 5930496, 30178575, 143184000, 639249300, 2705114880, 10914317934, 42189811200, 156883829400, 563116739584, 1956790259235, 6599620022400, 21651325216200, 69228721526400, 216108718571250, 659641645039360, 1971466420726656

Offset

-1,2

Comments

Euler transform of period 1 sequence [24,24,...].

Equals A023021 convolved with A000041. - Gary W. Adamson, Jun 09 2009

Equals convolution square of A005758: (1, 12, 90, 520, 2535, 10908, ...). - Gary W. Adamson, Jun 13 2009

Note the remarkably wide range of subjects where this sequence appears. - N. J. A. Sloane, Oct 29 2019

References

Arnaud Beauville, Counting rational curves on K3 surfaces, arXiv:alg-geom/9701019, Jan 1997.

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).

Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table III.

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.

C. L. Siegel, Advanced Analytic Number Theory, Tata Institute of Fundamental Research, Bombay, 1980, pp. 249-268.

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

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.

S.-T. YAU, E. ZASLOW: BPS states, string duality, and nodal curves on K3. Preprint arXiv:hep-th/9512121, 1995.

Links

Seiichi Manyama, Table of n, a(n) for n = -1..10000 (first 202 terms from T. D. Noe)

R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.

Richard E. Borcherds, Vinberg’s Algorithm and Kac-Moody algebras, Vinberg Lecture, Feb 26 2024.

Reinhold W. Gebert, Introduction to vertex algebras, Borcherds algebras and the Monster Lie algebra, 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]

Vaclav Kotesovec, Graph - the asymptotic ratio

Index entries for expansions of Product_{k >= 1} (1-x^k)^m

Formula

G.f.: (1/x)(Product_{k>0} (1-x^k))^-24 = 1/Delta (the discriminant in Siegel's notation).

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

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

Example

T_{14} = 1/q + 24 + 324q + 3200q^2 + 25650q^3 + ....

Maple

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

Mathematica

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 *)
CoefficientList[1/QPochhammer[q]^24 + O[q]^40, q] (* Jean-François Alcover, Nov 15 2015 *)

Prog

(PARI) a(n)=if(n<-1, 0, n++; polcoeff(eta(x+x*O(x^n))^-24, n))
(Julia) # DedekindEta is defined in A000594.
A006922List(len) = DedekindEta(len, -24)
A006922List(33) |> println # Peter Luschny, Mar 10 2018

Crossrefs

Cf. A000594, A048057, A048100, A048101, A048110, A048145.

24th column of A144064. - Alois P. Heinz, Oct 17 2008

Cf. A005758, A023021, A000041. - Gary W. Adamson, Jun 09 2009

Sequence in context: A239793 A289706 A300846 * A036221 A022652 A292298

Adjacent sequences: A006919 A006920 A006921 * A006923 A006924 A006925

Keyword

nonn,easy,nice

Author

N. J. A. Sloane

Extensions

More terms from Barry Brent (barryb(AT)primenet.com)

Status

approved