A004675 -id:A004675 - cp's OEIS frontend

A034597 Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.

1, 196560, 52416000, 6218175600, 565866362880, 45792819072000, 3486157968384000, 256206274225902000, 18422726047165440000, 1305984407917646096640, 91692325887531393024000

Offset: 0

N. J. A. Sloane

nonn

			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. A034598 (second coefficient, which eventually becomes negative), A034414, A034415.

# Extremal theta series:
with(numtheory): B := 1:
# set mu:
for mu from 1 to 10 do
   # set max deg:
   md := mu+3;
   f := 1+240*add(sigma[3](i)*x^i, i=1..md);
   f := series(f, x, md);
   f := series(f^3, x, md);
   g := series(x*mul((1-x^i)^24, i=1..md), x, md);
   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:
   B := B, coeff(A,x,mu+1);
od:
lprint(B);

terms = 11; Reap[For[mu = 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+1}]; Print[an]; Sow[an]]][[2,1]] (* Jean-François Alcover, Jul 08 2017, adapted from Maple *)

A018236 Weight distribution of hypothetical [ 72,36,16 ] doubly-even binary self-dual code.

Original entry on oeis.org

1, 0, 0, 0, 249849, 18106704, 462962955, 4397342400, 16602715899, 25756721120, 16602715899, 4397342400, 462962955, 18106704, 249849, 0, 0, 0, 1

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, Is There a (72,36) d = 16 Self-Dual Code?, IEEE Trans. Information Theory, vol. IT-19 (1973), p. 251.

E. M. Rains and N. J. A. Sloane, Self-dual codes, pp. 177-294 of Handbook of Coding Theory, Elsevier, 1998 (Abstract, pdf, ps).

Cf. A004675, A019590, A120373.

Let f = x^8 + 14 x^4 y^4 + y^8, g = x^4 y^4 (x^4-y^4)^4. Form the unique linear combination of f^9, f^6 g, f^3 g^2 and g^3 that begins x^72 + A_4 x^68 y^4 + A_8 x^64 y^8 + ..., with A_4 = A_8 = A_12 = 0, Set x=1, replace y^4 by y, and we have the g.f. for this sequence.

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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A034597 Leading coefficient of extremal theta series of even unimodular lattice in dimension 24n.

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A018236 Weight distribution of hypothetical [ 72,36,16 ] doubly-even binary self-dual code.

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

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