A115068 Triangle read by rows: T(n,k) = number of elements in the Coxeter group D_n with descent set contained in {s_k}, for 0<=k<=n-1.

1, 2, 2, 4, 6, 3, 8, 16, 12, 4, 16, 40, 40, 20, 5, 32, 96, 120, 80, 30, 6, 64, 224, 336, 280, 140, 42, 7, 128, 512, 896, 896, 560, 224, 56, 8, 256, 1152, 2304, 2688, 2016, 1008, 336, 72, 9, 512, 2560, 5760, 7680, 6720, 4032, 1680, 480, 90, 10, 1024, 5632, 14080, 21120

Offset

1,2

Comments

A115068 is the fission of the polynomial sequence (p(x,n)) by the polynomial sequence ((2x+1)^n), where p(n,x)=x^n+x^(n-1)+...+x+1, n>=0. See A193842 for the definition of fission. - Clark Kimberling, Aug 07 2011

References

A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, New York, 2005.

J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.

Links

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened

Formula

T(n,k)=binomial(n,k)*2^(n-k-1).

T(n,1) = 2^(n-1), T(n,n) = n, for n > 1: T(n,k) = T(n-1,k-1) + 2*T(n-1,k), 1 < k < n. - Reinhard Zumkeller, Jul 22 2013

Example

First six rows:
1
2...2
4...6....3
8...16...12...4
16..40...40...20...5
32..96...120..80...30...6

Mathematica

z = 11;
p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + 1;
q[n_, x_] := (2 x + 1)^n;
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x -> 0;
d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A115068 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, -1, z}]]   (* A193862 *)

Prog

(Haskell)
a115068 n k = a115068_tabl !! (n-1) !! (k-1)
a115068_row n = a115068_tabl !! (n-1)
a115068_tabl = iterate (\row -> zipWith (+) (row ++ [1]) $
                                zipWith (+) (row ++ [0]) ([0] ++ row)) [1]
-- Reinhard Zumkeller, Jul 22 2013

Crossrefs

Cf. A007318, A038207, A193862.

Cf. A105728, A013609.

Sequence in context: A260095 A209999 A127718 * A051495 A368157 A286542

Adjacent sequences: A115065 A115066 A115067 * A115069 A115070 A115071

Keyword

easy,nonn,tabl

Author

Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006

Status

approved