A193740 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=q(n,x) in A193738.

1, 1, 1, 1, 3, 3, 1, 4, 9, 9, 1, 4, 10, 19, 19, 1, 4, 10, 20, 34, 34, 1, 4, 10, 20, 35, 55, 55, 1, 4, 10, 20, 35, 56, 83, 83, 1, 4, 10, 20, 35, 56, 84, 119, 119, 1, 4, 10, 20, 35, 56, 84, 120, 164, 164, 1, 4, 10, 20, 35, 56, 84, 120, 165, 219, 219, 1, 4, 10, 20, 35, 56

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....3....3
1....4....9....9
1....4....10....19...19
1....4....10....20...34...34
		

Crossrefs

Cf. A193722, A193741.

Programs

Mathematica

z = 12;
p[0, x_] := 1
p[n_, x_] := n + Sum[(k + 1) x^(n - k), {k, 0, n - 1}]
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}]]  (* A193740 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]   (* A193741 *)