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
KEYWORD
AUTHOR
Elizabeth Morris (epmorris(AT)math.washington.edu), Mar 01 2006
STATUS
approved