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Triangle read by rows: row n gives coefficients of Schur polynomial Omega(n) in order of decreasing powers of x.
6

%I #72 Sep 12 2018 11:01:25

%S 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,

%T 26,73,152,222,203,8,1,7,34,111,275,511,703,623,13,1,8,43,159,452,997,

%U 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

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

%C Row n contains n+1 terms.

%H Gheorghe Coserea, <a href="/A269750/b269750.txt">Rows n = 0..200, flattened</a>

%H Andrew Misseldine, <a href="http://arxiv.org/abs/1508.03757">Counting Schur Rings over Cyclic Groups</a>, arXiv preprint arXiv:1508.03757 [math.RA], 2015.

%F 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

%e A(x) = 1 + t*x + (t^2 + t + 1)*x^2 + (t^3 + 2*t^2 + 4*t + 1)*x^3 + ...

%e Triangle begins:

%e n\k [0] [1] [2] [3] [4] [5] [6] [7] [8] [9]

%e [0] 1;

%e [1] 1, 0;

%e [2] 1, 1, 1;

%e [3] 1, 2, 4, 1;

%e [4] 1, 3, 8, 9, 2;

%e [5] 1, 4, 13, 23, 25, 3;

%e [6] 1, 5, 19, 44, 72, 69, 5;

%e [7] 1, 6, 26, 73, 152, 222, 203, 8;

%e [8] 1, 7, 34, 111, 275, 511, 703, 623, 13;

%e [9] 1, 8, 43, 159, 452, 997, 1725, 2272, 1990, 21;

%e [10]...

%t c[k_] := Binomial[2k, k]/(k+1);

%t 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}];

%t row[n_] := CoefficientList[om[n], x] // Reverse;

%t Table[row[n], {n, 0, 11}] // Flatten (* _Jean-François Alcover_, Sep 06 2018 *)

%o (PARI)

%o seq(N, t='t) = {

%o my(a=vector(N), c(k)=binomial(2*k, k)/(k+1)); a[1]=1; a[2]=t;

%o for (n = 2, N-1,

%o a[n+1] = t*a[n] + sum(k = 2, n, (c(k-1)*t+1)*a[n+1-k]));

%o return(a);

%o };

%o concat(apply(Vec, seq(12)))

%o (PARI)

%o N=12; x='x + O('x^N); t='t;

%o concat(apply(Vec, Vec(2*(1-x)/(-2*x^2 + (t-2)*(x-1) + t*(1-x)*sqrt(1-4*x)))))

%o \\ _Gheorghe Coserea_, Sep 10 2018

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

%Y 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).

%K nonn,tabl

%O 0,8

%A _N. J. A. Sloane_, Mar 22 2016

%E More terms from _Gheorghe Coserea_, Mar 24 2016