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

1,4

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

Links

Table of n, a(n) for n=1..55.

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

Example

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, ...
  ... (End)

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 *)

Crossrefs

Cf. A045623, A128176, A007318.

Sequence in context: A329854 A124725 A106522 * A104040 A338131 A332601

Adjacent sequences: A128172 A128173 A128174 * A128176 A128177 A128178

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 17 2007

Status

approved