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A269750 Triangle read by rows: row n gives coefficients of Schur polynomial Omega(n) in order of decreasing powers of x. 6
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A194524 A117136 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

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Last modified May 15 02:48 EDT 2021. Contains 343909 sequences. (Running on oeis4.)