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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.

%I A115068 #12 Mar 06 2014 22:33:18
%S A115068 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,
%T A115068 140,42,7,128,512,896,896,560,224,56,8,256,1152,2304,2688,2016,1008,
%U A115068 336,72,9,512,2560,5760,7680,6720,4032,1680,480,90,10,1024,5632,14080,21120
%N 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.
%C A115068 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
%D A115068 A. Bjorner and F. Brenti, Combinatorics of Coxeter Groups, Springer, New York, 2005.
%D A115068 J. E. Humphreys, Reflection Groups and Coxeter Groups, Cambridge University Press, Cambridge, 1990.
%H A115068 Reinhard Zumkeller, <a href="/A115068/b115068.txt">Rows n = 1..120 of triangle, flattened</a>
%F A115068 T(n,k)=binomial(n,k)*2^(n-k-1).
%F A115068 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
%e A115068 First six rows:
%e A115068 1
%e A115068 2...2
%e A115068 4...6....3
%e A115068 8...16...12...4
%e A115068 16..40...40...20...5
%e A115068 32..96...120..80...30...6
%t A115068 z = 11;
%t A115068 p[0, x_] := 1; p[n_, x_] := x*p[n - 1, x] + 1;
%t A115068 q[n_, x_] := (2 x + 1)^n;
%t A115068 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A115068 p1[n_, 0] := p[n, x] /. x -> 0;
%t A115068 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A115068 h[n_] := CoefficientList[d[n, x], {x}]
%t A115068 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A115068 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A115068 *)
%t A115068 TableForm[Table[h[n], {n, 0, z}]]
%t A115068 Flatten[Table[h[n], {n, -1, z}]]   (* A193862 *)
%o A115068 (Haskell)
%o A115068 a115068 n k = a115068_tabl !! (n-1) !! (k-1)
%o A115068 a115068_row n = a115068_tabl !! (n-1)
%o A115068 a115068_tabl = iterate (\row -> zipWith (+) (row ++ [1]) $
%o A115068                                 zipWith (+) (row ++ [0]) ([0] ++ row)) [1]
%o A115068 -- _Reinhard Zumkeller_, Jul 22 2013
%Y A115068 Cf. A007318, A038207, A193862.
%Y A115068 Cf. A105728, A013609.
%K A115068 easy,nonn,tabl
%O A115068 1,2
%A A115068 Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006