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