A128175 Binomial transform of A128174.

1, 1, 1, 2, 2, 1, 4, 4, 3, 1, 8, 8, 7, 4, 1, 16, 16, 15, 11, 5, 1, 32, 32, 31, 26, 16, 6, 1, 64, 64, 63, 57, 42, 22, 7, 1, 128, 128, 127, 120, 99, 64, 29, 8, 1, 256, 256, 255, 247, 219, 163, 93, 37, 9, 1

Offset: 1

Author

Gary W. Adamson, Feb 17 2007

Keywords

• nonn
• tabl

Comments

Row sums = A045623: (1, 2, 5, 12, 28, 64, 144, ...).

A128176 = A128174 * A007318.

Riordan array ((1-x)/(1-2x),x/(1-x)). - Paul Barry, Oct 02 2010

Fusion of polynomial sequences p(n,x) = (x+1)^n and q(n,x) = x^n + x^(n-1) + ... + x + 1; see A193722 for the definition of fusion. - Clark Kimberling, Aug 04 2011

Examples

			First few rows of the triangle:
   1;
   1,  1;
   2,  2,  1;
   4,  4,  3,  1;
   8,  8,  7,  4,  1;
  16, 16, 15, 11,  5,  1;
  32, 32, 31, 26, 16,  6,  1;
  64, 64, 63, 57, 42, 22,  7,  1;
  ...
From _Paul Barry_, Oct 02 2010: (Start)
Production matrix is
  1, 1;
  1, 1, 1;
  0, 0, 1, 1;
  0, 0, 0, 1, 1;
  0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 1, 1;
  0, 0, 0, 0, 0, 0, 0, 0, 1, 1;
  ...
Matrix logarithm is
  0;
  1, 0;
  1, 2, 0;
  1, 1, 3, 0;
  1, 1, 1, 4, 0;
  1, 1, 1, 1, 5, 0;
  1, 1, 1, 1, 1, 6, 0;
  1, 1, 1, 1, 1, 1, 7, 0;
  1, 1, 1, 1, 1, 1, 1, 8, 0;
  1, 1, 1, 1, 1, 1, 1, 1, 9,  0;
  1, 1, 1, 1, 1, 1, 1, 1, 1, 10, 0;
  ... (End)
.
First few rows of the array:
  1, 1,  2,  4,  8,  16, ...
  1, 2,  4,  8, 16,  32, ...
  1, 3,  7, 15, 31,  63, ...
  1, 4, 11, 26, 57, 120, ...
  1, 5, 16, 42, 99, 219, ...
  ...
		

Crossrefs

Cf. A045623, A128176, A007318.

Programs

Maple

A193820 := (n,k) -> `if`(k=0 or n=0, 1, A193820(n-1,k-1)+A193820(n-1,k));
A128175 := (n,k) -> A193820(n-1,n-k);
seq(print(seq(A128175(n,k),k=0..n)),n=0..10); # Peter Luschny, Jan 22 2012

Mathematica

z = 10; a = 1; b = 1;
p[n_, x_] := (a*x + b)^n
q[0, x_] := 1
q[n_, x_] := x*q[n - 1, x] + 1; q[n_, 0] := q[n, x] /. x -> 0;
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}]]   (* A193820 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A128175 *)
(* Clark Kimberling, Aug 06 2011 *)
(* function dotTriangle[] is defined in A128176 *)
a128175[r_] := dotTriangle[Binomial, If[EvenQ[#1 + #2], 1, 0]&, r]
TableForm[a128174[7]] (* triangle *)
Flatten[a128174[9]] (* data *) (* Hartmut F. W. Hoft, Mar 15 2017 *)

Formula

A007318 * A128174 as infinite lower triangular matrices.

Antidiagonals of an array in which the first row = (1, 1, 2, 4, 8, 16, ...); and (n+1)-th row = partial sums of n-th row.

exp(x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(x)*(4 + 4*x + 3*x^2/2! + x^3/3!) = 4 + 8*x + 15*x^2/2! + 26*x^3/3! + 42*x^4/4! + .... The same property holds more generally for Riordan arrays of the form ( f(x), x/(1 - x) ). - Peter Bala, Dec 22 2014

T(n, k) = Sum_{i=0..floor((n-k)/2)} binomial(n-1, k-1+2*i). - Werner Schulte, Mar 05 2025

T(n, k) = binomial(n-1, k-1)*hypergeom([1, (k-n)/2, (1+k-n)/2], [(1+k)/2, k/2], 1). - Stefano Spezia, Mar 07 2025