A193738 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=q(n,x)=x^n+x^(n-1)+...+x+1.

1, 1, 1, 1, 2, 2, 1, 2, 3, 3, 1, 2, 3, 4, 4, 1, 2, 3, 4, 5, 5, 1, 2, 3, 4, 5, 6, 6, 1, 2, 3, 4, 5, 6, 7, 7, 1, 2, 3, 4, 5, 6, 7, 8, 8, 1, 2, 3, 4, 5, 6, 7, 8, 9, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 10, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 11, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 12, 1, 2, 3, 4

Offset: 0

Author

Clark Kimberling, Aug 04 2011

Keywords

• nonn
• tabl

Comments

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

Examples

			First six rows:
1
1....1
1....2....2
1....2....3....3
1....2....3....4...4
1....2....3....4...5...5
		

Links

• Reinhard Zumkeller, Rows n = 0..100 of triangle, flattened

Crossrefs

Cf. A193722, A193739.

Programs

Haskell

a193738 n k = a193738_tabl !! n !! k
a193738_row n = a193738_tabl !! n
a193738_tabl = map reverse a193739_tabl
-- Reinhard Zumkeller, May 11 2013

Mathematica

z = 12;
p[0, x_] := 1
p[n_, x_] := x*p[n - 1, x] + 1; p[n_, 0] := p[n, x] /. x -> 0
q[n_, x_] := p[n, x]
t[n_, k_] := Coefficient[p[n, x], x^(n - k)];
t[n_, n_] := 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}]]  (* A193738 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193739 *)