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)
G.f.: x*(1 - 9*x + 24*x^2 - 19*x^3)/(1 - 10*x + 32*x^2 - 37*x^3 + 11*x^4). - Charles R Greathouse IV, May 20 2026
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
(PARI) a(n)=([0, 1, 0, 0; 0, 0, 1, 0; 0, 0, 0, 1; -11, 37, -32, 10]^(n-1)*[1; 1; 2; 6])[1, 1] \\ Charles R Greathouse IV, May 20 2026
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Mike Zabrocki, Oct 24 2006
STATUS
approved