A270786 Number of Schur rings over Z_{5^n}.

1, 3, 13, 58, 263, 1203, 5531, 25511, 117910, 545730, 2528263, 11720917, 54364253, 252243996, 1170694877, 5434421231, 25230661483, 117153235821, 544024844668, 2526465046405, 11733602605442, 54496414141998, 253115681845187, 1175659969364675, 5460766440081739

Offset

0,2

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.

Formula

G.f.: 2*(1-x)/(-2*x^2 + (x-1) + 3*(1-x)*sqrt(1-4*x)); equivalently, the g.f. can be rewritten as -y^2*(y^2 - y + 1)/(2*y^4 - 5*y^3 + 6*y^2 - 5*y + 1), where y=A000108(x). - Gheorghe Coserea, Sep 10 2018

Mathematica

om[n_] := om[n] = x om[n - 1] + Sum[(CatalanNumber[k - 1] x + 1) om[n - k], {k, 2, n}] // Expand; om[0] = 1; om[1] = x;
Array[om, 25, 0] /. x -> 3 (* Jean-François Alcover, Oct 25 2018 *)

Prog

(PARI)
A269750_seq(N, t='t) = {
  my(a=vector(N), c(k)=binomial(2*k, k)/(k+1)); a[1]=1; a[2]=t;
  for (n = 2, N-1,
    a[n+1] = t*a[n] + sum(k = 2, n, (c(k-1)*t+1)*a[n+1-k]));
  return(a);
};
A269750_seq(25, numdiv(4)) \\ Gheorghe Coserea, Sep 10 2018

Crossrefs

Cf. A000108, A269750.

Sequence in context: A151225 A326984 A296771 * A151226 A151320 A151227

Adjacent sequences: A270783 A270784 A270785 * A270787 A270788 A270789

Keyword

nonn

Author

N. J. A. Sloane, Mar 23 2016

Extensions

More terms from Gheorghe Coserea, Mar 24 2016

a(0)=1 prepended by Gheorghe Coserea, Sep 10 2018

Status

approved