A034597 -id:A034597 - cp's OEIS frontend

A008408 Theta series of Leech lattice.

1, 0, 196560, 16773120, 398034000, 4629381120, 34417656000, 187489935360, 814879774800, 2975551488000, 9486551299680, 27052945920000, 70486236999360, 169931095326720, 384163586352000, 820166620815360, 1668890090322000

Offset: 0

N. J. A. Sloane

			G.f. = 1 + 196560*q^2 + 16773120*q^3 + 398034000*q^4 + 4629381120*q^5 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Third Edition, Springer-Verlag,1993, pp. 51, 134-135.
W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 113.
N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (first 501 terms from N. J. A. Sloane)
Henry Cohn, Abhinav Kumar, Stephen D. Miller, Danylo Radchenko, and Maryna Viazovska, The sphere packing problem in dimension 24, arXiv:1603.06518 [math.NT], 2016.
Henry Cohn and Stephen D. Miller, Some properties of optimal functions for sphere packing in dimensions 8 and 24, arXiv:1603.04759 [math.MG], 2016
David de Laat and Frank Vallentin, A Breakthrough in Sphere Packing: The Search for Magic Functions, arXiv preprint arXiv:1607.02111 [math.MG], 2016.
Fern Gossow, Lyndon-like cyclic sieving and Gauss congruence, arXiv:2410.05678 [math.CO], 2024. See p. 26.
Nadia Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006; J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
G. Nebe and N. J. A. Sloane, Home page for lattice
K. Ono, S. Robins and P. T. Wahl, On the Representation of Integers as Sums of Triangular Numbers, (see p. 12), Aequationes mathematicae, August 1995, Volume 50, Issue 1-2, pp 73-94.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane, Seven Staggering Sequences.
Eric Weisstein's World of Mathematics, Leech Lattice.
Eric Weisstein's World of Mathematics, Theta Series.

Cf. A004009, A108093, A000594, A108093 (24th root), A034597, A034598, A198343, A013959.

// Theta series of the Leech lattice, from John Cannon, Dec 29 2006
A008408Q := function(prec) M12 := ModularForms(Gamma0(1), 12); t1 := Basis(M12)[1]; T := PowerSeries(t1, prec); return Coefficients(T); end function; Q := A008408Q(1000); Q[678];

Basis( ModularForms( Gamma0(1), 12), 30) [1] ; /* Michael Somos, Jun 09 2014 */

with(numtheory); f := 1+240*add(sigma[ 3 ](m)*q^(2*m),m=1..50); t := q^2*mul((1-q^(2*m))^24,m=1..50); series(f^3-720*t,q,51);

max = 17; f = 1 + 240*Sum[ DivisorSigma[3, m]*q^(2m), {m, 1, max}]; t = q^2*Product[(1 - q^(2m))^24, {m, 1, max}]; Partition[ CoefficientList[ Series[f^3 - 720t, {q, 0, 2 max}], q], 2][[All, 1]] (* Jean-François Alcover , Oct 14 2011, after Maple *)
(* From version 6 on *) f[q_] = LatticeData["Leech", "ThetaSeriesFunction"][x] /. x -> -I*Log[q]/Pi; Series[f[q], {q, 0, 32}] // CoefficientList[#, q^2]& (* Jean-François Alcover, May 15 2013 *)
a[ n_] := If[ n < 1, Boole[n == 0], SeriesCoefficient[(1 + 240 Sum[ q^k DivisorSigma[ 3, k], {k, n}])^3 - 720 q QPochhammer[ q]^24, {q, 0, n}]]; (* Michael Somos, Jun 09 2014 *)

{a(n) = if( n<1, n==0, polcoeff( 1 + (sum(k=1, n, sigma(k,11)*x^k) - x*eta(x + O(x^n))^24) * 65520/691, n))}; /* Michael Somos, Oct 19 2006 */

{a(n) = if( n<1, n==0, polcoeff( sum(k=1, n, 240*sigma(k,3)*x^k, 1 + x*O(x^n))^3 - 720*x*eta(x + O(x^n))^24, n))}; /* Michael Somos, Oct 19 2006 */

from sympy import divisor_sigma
def A008408(n): return 65520*(divisor_sigma(n,11)-(n**4*divisor_sigma(n)-24*((m:=n+1>>1)**2*(0 if n&1 else (m*(35*m - 52*n) + 18*n**2)*divisor_sigma(m)**2)+sum((i*(i*(i*(70*i - 140*n) + 90*n**2) - 20*n**3) + n**4)*divisor_sigma(i)*divisor_sigma(n-i) for i in range(1,m)))))//691 if n else 1 # Chai Wah Wu, Nov 17 2022

A = ModularForms( Gamma0(1), 12, prec=30) . basis() ; A[1] - 65520/691*A[0] # Michael Somos, Jun 09 2014

The simplest way to obtain this is to take the cube of the theta series for E_8 (A004009) and subtract 720 times the g.f. for the Ramanujan numbers (A000594).
This theta series is thus also the q-expansion of (7/12) E_4(z)^3 + (5/12) E_6(z)^2. Cf. A013973. - Daniel D. Briggs, Nov 25 2011
a(n) = 65520*(A013959(n) - A000594(n))/691, n >= 1. a(0) = 1. Expansion of the Theta series of the Leech lattice in powers of q^2. See the Conway and Sloane reference. - Wolfdieter Lang, Jan 16 2017

A034414 Leading term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.

Original entry on oeis.org

1, 759, 17296, 249849, 3217056, 39703755, 481008528, 5776211364, 69065734464, 824142912363, 9826364199840, 117145945726810, 1396918583188128, 16665451879695801, 198937019774252928

Offset: 0

N. J. A. Sloane

The term after the leading nonzero term eventually becomes negative and so for large n the extremal codes do not exist (see references, also A034415).

			At length 24, the extremal weight enumerator is 1+759*x^8+2576*x^12+..., with leading coefficient 759; this is the weight enumerator of the binary Golay code.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, see Theorem 13, p. 624.
C. L. Mallows and N. J. A. Sloane, An Upper Bound for Self-Dual Codes, Information and Control, 22 (1973), 188-200.

Seiichi Manyama, Table of n, a(n) for n = 0..919 (terms 0..250 from N. J. A. Sloane)
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A034415 (second coefficient, which becmes negative), A001380, A034597.

# Extremal weight enumerators:
kernelopts(printbytes=false): interface(screenwidth=200);
W0:=1; f:=1+14*x+x^2; f:=f^3; g:=x*(1-x)^4;
for mu from 1 to 100 do
# set max deg
md:=mu+3; W0:=series(f^mu,x,md): h:=series(g/f,x,md): A:=series(W0,x,md): Z:=A:
for i from 1 to mu do
Z:=series(Z*h,x,md); A:=series(A-coeff(A,x,i)*Z,x,md); od: lprint(A);
od:

a[n_] := 18(6n-1)(8n-1)(12n-1)(24n-1)Binomial[5n-2, n-1]/((n+1)(2n+1)(4n+1)(4n+3)); a[0] = 1; Table[a[n], {n, 0, 14}](* Jean-François Alcover, Oct 06 2011, after formula *)

a(24n) = C(24n, 5)*C(5n-2, n-1)/C(4n+4, 5).

A034415 Second term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.

Original entry on oeis.org

1, 2576, 535095, 18106704, 369844880, 6101289120, 90184804281, 1251098739072, 16681003659936, 216644275600560, 2763033644875595, 34784314216176096, 433742858109499536, 5369839142579042560

Offset: 0

N. J. A. Sloane

sign

The terms become negative at n=154 and so certainly by that point the extremal codes do not exist (see references).

Up to n = 250 the terms steadily increase in magnitude, but their sign changes from positive to negative at n = 154.

			At length 24, the weight enumerator (of the Golay code) is 1+759*x^8+2576*x^12+..., with leading coefficient 759 and second term 2576.

F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, see Theorem 13, p. 624.

N. J. A. Sloane, Table of n, a(n) for n = 0..250
C. L. Mallows and N. J. A. Sloane, An Upper Bound for Self-Dual Codes, Information and Control, 22 (1973), 188-200.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A034414 (leading coefficient), A001380, A034597, A034598.

Maple
```
For Maple program see A034414.
```

A034598 Second coefficient of extremal theta series of even unimodular lattice in dimension 24n.

Original entry on oeis.org

1, 16773120, 39007332000, 15281788354560, 2972108280960000, 406954241261568000, 45569082381053868000, 4499117081888292864000, 408472720963469499617280, 34975479259332252426240000

Offset: 0

N. J. A. Sloane

sign

Although these initially increase, they eventually go negative at about term 1700 (i.e. dimension about 40800) - see references.

			When n=1 we get the theta series of the 24-dimensional Leech lattice: 1+196560*q^4+16773120*q^6+... (see A008408). For n=2 we get A004672 and for n=3, A004675.

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.

N. J. A. Sloane, Table of n, a(n) for n = 0..100
C. L. Mallows, A. M. Odlyzko and N. J. A. Sloane, Upper bounds for modular forms, lattices and codes, J. Alg., 36 (1975), 68-76.
C. L. Mallows and N. J. A. Sloane, An Upper Bound for Self-Dual Codes, Information and Control, 22 (1973), 188-200.
G. Nebe, E. M. Rains and N. J. A. Sloane, Self-Dual Codes and Invariant Theory, Springer, Berlin, 2006.
E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).

Cf. A034597 (leading coefficient).

Maple
```
For Maple program see A034597.
```

terms = 10; Reap[For[mu = 1; Print[1]; Sow[1], mu < terms, mu++, md = mu + 3; f = 1 + 240*Sum[DivisorSigma[3, i]*x^i, {i, 1, md}]; f = Series[f, {x, 0, md}]; f = Series[f^3, {x, 0, md}]; g = Series[x*Product[ (1 - x^i)^24, {i, 1, md}], {x, 0, md}]; W0 = Series[f^mu, {x, 0, md}]; h = Series[g/f, {x, 0, md}]; A = Series[W0, {x, 0, md}]; Z = A; For[i = 1, i <= mu, i++, Z = Series[Z*h, {x, 0, md}]; A = Series[A - SeriesCoefficient[A, {x, 0, i}]*Z, {x, 0, md}]]; an = SeriesCoefficient[A, {x, 0, mu+2}]; Print[an]; Sow[an]]][[2,1]] (* Jean-François Alcover, Jul 08 2017, adapted from Maple program for A034597 *)

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A008408 Theta series of Leech lattice.

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A034414 Leading term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.

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A034415 Second term in extremal weight enumerator of doubly-even binary self-dual code of length 24n.

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A034598 Second coefficient of extremal theta series of even unimodular lattice in dimension 24n.

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