cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A001380 Weight distribution of binary Golay code of length 24.

Original entry on oeis.org

1, 0, 759, 2576, 759, 0, 1
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			The weight enumerator is x^24+759*x^16*y^8+2576*x^12*y^12+759*x^8*y^16+y^24.

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 84.
  • W. Ebeling, Lattices and Codes, Vieweg; 2nd ed., 2002, see p. 71.
  • F. J. MacWilliams and N. J. A. Sloane, The Theory of Error-Correcting Codes, Elsevier-North Holland, 1978, p. 67.

Links

Crossrefs

Cf. A002289, A034414, A034415.

Programs

  • Maple
    g24 := x^24+759*x^16*y^8+759*x^8*y^16+2576*x^12*y^12+y^24; e8 := x^8+14*x^4*y^4+y^8; d:=n->x^(n mod 2)*(1/2)*( (x^2+y^2)^floor((n)/2)+(x^2-y^2)^floor((n)/2));

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

Original entry on oeis.org

1, 196560, 52416000, 6218175600, 565866362880, 45792819072000, 3486157968384000, 256206274225902000, 18422726047165440000, 1305984407917646096640, 91692325887531393024000
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Examples

			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.

References

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

Links

Crossrefs

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

Programs

  • Maple
    # 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);
  • Mathematica
    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 *)

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

Views

Author

N. J. A. Sloane

Keywords

Comments

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).

Examples

			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.

References

  • 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.

Links

Crossrefs

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

Programs

  • Maple
    # 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:
  • Mathematica
    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 *)

Formula

a(24n) = C(24n, 5)*C(5n-2, n-1)/C(4n+4, 5).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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