A270789 Number of Schur rings over Z_{2^n}.

1, 1, 3, 10, 37, 151, 657, 2989, 14044, 67626, 332061, 1656893, 8379773, 42871890, 221528317, 1154624725, 6063752579, 32058277129, 170492969062, 911505351447, 4896187116904, 26411691702030, 143019573086693, 777143219649275, 4236216759058815

Offset: 0

Author

N. J. A. Sloane, Mar 23 2016

Keywords

• nonn

Links

• Gheorghe Coserea, Table of n, a(n) for n = 0..1001
• Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015.

Crossrefs

Cf. A000108, A006318, A269750, A270785, A270786, A270787.

Programs

Mathematica

A000108[x_] = (1 - Sqrt[1 - 4x])/(2x);
A006318[x_] = (1 - x - (1 - 6x + x^2)^(1/2))/(2x);
y = A000108[x] + A006318[x] - 1;
gf = (x-1) (y x-1) / (y x (x^2 + x - 1) - 2x + 1);
CoefficientList[gf + O[x]^25, x] (* Jean-François Alcover, Oct 06 2018, after Gheorghe Coserea *)

PARI

seq(N) = {
  my(a=vector(N), sv=vector(N), cs=vector(N), c(k)=binomial(2*k, k)/(k+1),
     s(k) = sum(j=0, k, binomial(2*j, j)*binomial(k + j, 2*j)/(j+1)));
  a[1] = 1; a[2] = 1; a[3] = 3; a[4] = 10;
  sv[1] = s(1); cs[1] = c(1) + sv[1];
  for (j = 2, #cs, sv[j] = s(j); cs[j] = cs[j-1] + c(j) + sv[j]);
  for (n = 4, #a-1,
    a[n+1] = sum(k = 1, 3, 2^k * a[n+1-k]) - (c(n-1) + sv[n-1]) +
             sum(k = 4, n, (c(k-1) + sv[k-1] - cs[k-3]) * a[n+1-k]));
  return(a);
};
seq(25)

PARI

seq(N) = {
  my(x='x+O('x^N), c=(1-(1-4*x)^(1/2))/(2*x),
     s=(1-x-(1-6*x+x^2)^(1/2))/(2*x), y=c+s-1);
  Vec((x - 1)*(y*x - 1)/(y*x*(x^2 + x - 1) - 2*x + 1));
};
seq(25) \\ Gheorghe Coserea, Sep 12 2018

Formula

G.f.: ((2-x-sqrt(1-4*x) - sqrt(1-6*x+x^2))*(1-x) + 2*(x^2-1))/((2-x-sqrt(1-4*x)-sqrt(1-6*x+x^2))*(1-x-x^2) + 2*(x^3+x^2+x-1)) (see Misseldine link); equivalently, the g.f. can be rewritten as (x - 1)*(y*x - 1)/(y*x*(x^2 + x - 1) - 2*x + 1), where y = A000108(x) + A006318(x) - 1. - Gheorghe Coserea, Sep 12 2018

Extensions

More terms from Gheorghe Coserea, Mar 24 2016

a(0)=1 prepended by Gheorghe Coserea, Sep 12 2018