OFFSET
0,4
COMMENTS
Row sums = A095121: (1, 2, 6, 14, 30, 62, 126,...).
Triangle T(n,k), 0<=k<=n, read by rows, given by [1,1,-1,1,0,0,0,0,0,0,0,...] DELTA [1,0,-1,1,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 03 2009
A153861 is the fusion of polynomial sequences p(n,x)=x^n+x^(n-1)+...+x+1 and q(n,x)=(x+1)^n; see A193722 for the definition of fusion. - Clark Kimberling, Aug 06 2011
LINKS
G. C. Greubel, Table of n, a(n) for the first 46 rows
FORMULA
Triangle read by rows, A007318 * A153860. Remove left two columns of Pascal's triangle and append (1, 1, 2, 3, 4, 5,...).
As a recursive operation by way of example, row 3 = (3, 6, 4, 1) =
[1, 1, 1, 0] * (flipped Pascal's triangle matrix) = [1, 3, 3, 1]
[1, 2, 1, 0]
[1, 1, 0, 0]
[1, 0, 0, 0].
T(n,k) = 2*T(n-1,k)+T(n-1,k-1)-T(n-2,k)-T(n-2,k-1), T(0,0) = T(1,0) = T(1,1) = T(2,2) = 1, T(2,0)=2, T(2,1)=3, T(n,k)=0 if k<0 or if k>n. - Philippe Deléham, Dec 15 2013
G.f.: (1-x+x^2+x^2*y)/((x-1)*(-1+x+x*y)). - R. J. Mathar, Aug 11 2015
EXAMPLE
First few rows of the triangle are:
1;
1, 1;
2, 3, 1;
3, 6, 4, 1;
4, 10, 10, 5, 1;
5, 15, 20, 15, 6, 1;
6, 21, 35, 35, 21, 7, 1;
7, 28, 56, 70, 56, 28, 8, 1;
8, 36, 84, 126, 126, 84, 36, 9, 1;
9, 45, 120, 210, 252, 210, 120, 45, 10, 1;
...
MATHEMATICA
z = 10; c = 1; d = 1;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0;
q[n_, x_] := (c*x + d)^n
t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
g[n_] := CoefficientList[w[n, x], {x}]
TableForm[Table[Reverse[g[n]], {n, -1, z}]]
Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193815 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]] (* A153861 *)
(* Clark Kimberling, Aug 06 2011 *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, Jan 03 2009
STATUS
approved