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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. 7
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 (list; table; graph; refs; listen; history; text; internal format)
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
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
Sequence in context: A260095 A209999 A127718 * A051495 A368157 A286542
KEYWORD
easy,nonn,tabl
AUTHOR
Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006
STATUS
approved

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Last modified March 19 01:57 EDT 2024. Contains 370952 sequences. (Running on oeis4.)