A193820 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=x^n+x^(n-1)+...+x+1.

1, 1, 1, 1, 2, 2, 1, 3, 4, 4, 1, 4, 7, 8, 8, 1, 5, 11, 15, 16, 16, 1, 6, 16, 26, 31, 32, 32, 1, 7, 22, 42, 57, 63, 64, 64, 1, 8, 29, 64, 99, 120, 127, 128, 128, 1, 9, 37, 93, 163, 219, 247, 255, 256, 256, 1, 10, 46, 130, 256, 382, 466, 502, 511, 512, 512, 1, 11, 56

Offset: 0

Author

Clark Kimberling, Aug 06 2011

Keywords

• nonn
• tabl

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays.

Variant of A054143 and A008949. - R. J. Mathar, Mar 03 2013

Examples

			First six rows:
  1
  1....1
  1....2....2
  1....3....4....4
  1....4....7....8....8
  1....5....11...15...16...16
		

Crossrefs

Cf. A193722, A128175, A193823, A045623 (row sums), A009766.

Cf. A054143, A008949.

Programs

Maple

A193820 := (n,k) -> `if`(k=0 or n=0,1, A193820(n-1,k-1)+A193820(n-1,k));
seq(print(seq(A193820(n,k),k=0..n+1)),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 *)

Formula

From Peter Bala, Jul 16 2013: (Start)

T(n,k) = sum {i = 0..k} binomial(n-1,k-i) for 0 <= k <= n.

O.g.f.: (1 - x*t)^2/( (1 - 2*x*t)*(1 - (1 + x)*t) ) = 1 + (1 + x)*t + (1 + 2*x + 2*x^2)*t^2 + ....

The n-th row polynomial R(n,x) for n >= 1 is given by R(n,x) = 1/(1 - x)*( (x + 1)^(n-1) - 2^(n-1)*x^(n+1) ). Cf. A193823. (End)