A269750 Triangle read by rows: row n gives coefficients of Schur polynomial Omega(n) in order of decreasing powers of x.

1, 1, 0, 1, 1, 1, 1, 2, 4, 1, 1, 3, 8, 9, 2, 1, 4, 13, 23, 25, 3, 1, 5, 19, 44, 72, 69, 5, 1, 6, 26, 73, 152, 222, 203, 8, 1, 7, 34, 111, 275, 511, 703, 623, 13, 1, 8, 43, 159, 452, 997, 1725, 2272, 1990, 21, 1, 9, 53, 218, 695, 1754, 3572, 5854, 7510, 6559, 34, 1, 10, 64, 289, 1017, 2870, 6645, 12717, 20065, 25325, 22161, 55

Offset

0,8

Comments

Row n contains n+1 terms.

Links

Gheorghe Coserea, Rows n = 0..200, flattened

Andrew Misseldine, Counting Schur Rings over Cyclic Groups, arXiv preprint arXiv:1508.03757 [math.RA], 2015.

Formula

G.f. A(x) = Sum_{n>=0} P_n(t)*x^n = 2*(1-x)/(-2*x^2 + (t-2)*(x-1) + t*(1-x)*sqrt(1-4*x)), where P_n(t) = Sum_{k=0..n} T(n,k)*t^(n-k) (see Misseldine link); equivalently, the g.f. can be rewritten as y^2*(y^2 - y + 1)/(y^4 - y^3 + 2*y - 1 - t*y*(y - 1)*(y^2 - y + 1)), where y=A000108(x). - Gheorghe Coserea, Sep 10 2018

Example

A(x) = 1 + t*x + (t^2 + t + 1)*x^2 + (t^3 + 2*t^2 + 4*t + 1)*x^3 + ...
Triangle begins:
n\k [0]    [1]    [2]    [3]    [4]    [5]    [6]    [7]    [8]    [9]
[0] 1;
[1] 1,     0;
[2] 1,     1,     1;
[3] 1,     2,     4,     1;
[4] 1,     3,     8,     9,     2;
[5] 1,     4,     13,    23,    25,    3;
[6] 1,     5,     19,    44,    72,    69,    5;
[7] 1,     6,     26,    73,    152,   222,   203,   8;
[8] 1,     7,     34,    111,   275,   511,   703,   623,   13;
[9] 1,     8,     43,    159,   452,   997,   1725,  2272,  1990,  21;
[10]...

Mathematica

c[k_] := Binomial[2k, k]/(k+1);
om[0] = 1; om[1] = x; om[n_] := om[n] = x om[n-1] + Sum[(c[k-1] x + 1) om[n - k], {k, 2, n}];
row[n_] := CoefficientList[om[n], x] // Reverse;
Table[row[n], {n, 0, 11}] // Flatten (* Jean-François Alcover, Sep 06 2018 *)

Prog

(PARI)
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);
};
concat(apply(Vec, seq(12)))
(PARI)
N=12; x='x + O('x^N); t='t;
concat(apply(Vec, Vec(2*(1-x)/(-2*x^2 + (t-2)*(x-1) + t*(1-x)*sqrt(1-4*x)))))
\\ Gheorghe Coserea, Sep 10 2018

Crossrefs

Cf. A000040, A000045(n-1)=P_n(0), A000108, A270789.

For odd prime p, evaluating the polynomial P_n(t) at t=A000005(p-1) gives the number of Schur rings over Z_{p^n}. For p=3,5,7 we have t=2,3,4 and the associated sequences A270785(n) = P_n(2), A270786(n) = P_n(3), A270787(n) = P_n(4).

Sequence in context: A117136 A362142 A139227 * A292495 A065626 A201758

Adjacent sequences: A269747 A269748 A269749 * A269751 A269752 A269753

Keyword

nonn,tabl

Author

N. J. A. Sloane, Mar 22 2016

Extensions

More terms from Gheorghe Coserea, Mar 24 2016

Status

approved