A124293 Number of free generators of degree n of symmetric polynomials in 5-noncommuting variables.

1, 1, 2, 6, 22, 91, 406, 1896, 9093, 44279, 217500, 1073657, 5314870, 26352107, 130778039, 649352929, 3225196431, 16021584848, 79597062632, 395469296912, 1964908443531, 9762920818182, 48508934285620, 241027326818991, 1197601448443963, 5950578465799856

Offset

1,3

Comments

Also the number of non-splitable set partitions (see Bergeron et al. reference) of length <=5

Links

Alois P. Heinz, Table of n, a(n) for n = 1..500

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, arXiv:math/0502082 [math.CO], 2005.

N. Bergeron, C. Reutenauer, M. Rosas and M. Zabrocki, Invariants and Coinvariants of the Symmetric Group in Noncommuting Variables, Canad. J. Math. 60 (2008), no. 2, 266-296.

M. C. Wolf, Symmetric functions of noncommutative elements, Duke Math. J. 2 (1936), 626-637.

Index entries for linear recurrences with constant coefficients, signature (10,-32,37,-11).

Formula

O.g.f. (1-9q+24q^2-19q^3)/(1-10q+32q^2-37q^3+11q^4) = (1 - 1/(sum_{k=0}^5 q^k/(prod_{i=1}^k (1-i*q))))/q a(n) = add( A055105(n,k), k=1..5) = add(A055106(n,k),k=1..4)

Maple

a:= n-> (Matrix([[6, 2, 1, 1]]). Matrix(4, (i, j)-> if i=j-1 then 1 elif j=1 then [10, -32, 37, -11][i] else 0 fi)^(n-1))[1, 4]: seq(a(n), n=1..30); # Alois P. Heinz, Sep 05 2008

Mathematica

LinearRecurrence[{10, -32, 37, -11}, {1, 1, 2, 6}, 30] (* Jean-François Alcover, Jan 08 2016 *)

Prog

(Magma) I:=[1, 1, 2, 6]; [n le 4 select I[n] else 10*Self(n-1)-32*Self(n-2)+37*Self(n-3)-11*Self(n-4): n in [1..30]]; // Vincenzo Librandi, Jan 09 2016

Crossrefs

Cf. A055105, A055106, A055107, A074664, A001519, A124292, A124294, A124295.

Sequence in context: A150271 A150272 A342289 * A341382 A107591 A279569

Adjacent sequences: A124290 A124291 A124292 * A124294 A124295 A124296

Keyword

nonn

Author

Mike Zabrocki, Oct 24 2006

Status

approved