A153861 Triangle read by rows, binomial transform of triangle A153860.

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, 10, 55, 165, 330, 462, 462, 330, 165, 55, 11, 1

Offset: 0

Author

Gary W. Adamson, Jan 03 2009

Keywords

• nonn
• tabl

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

A123110*A007318 as infinite lower triangular matrices. - Philippe Deléham, Jan 06 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

Examples

			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;
...
		

Links

• G. C. Greubel, Table of n, a(n) for the first 46 rows

Crossrefs

Cf. A153860, A095121, A130405.

This is A137396 without the initial column and without signs.

Programs

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 *)

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].

(Cf. analogous operation in A130405, but in A153861 the linear multiplier = [1,1,1,...,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