A193787 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(x+1)^n and q(n,x)=1+x^n.

1, 1, 1, 1, 1, 2, 1, 2, 1, 4, 1, 3, 3, 1, 8, 1, 4, 6, 4, 1, 16, 1, 5, 10, 10, 5, 1, 32, 1, 6, 15, 20, 15, 6, 1, 64, 1, 7, 21, 35, 35, 21, 7, 1, 128, 1, 8, 28, 56, 70, 56, 28, 8, 1, 256, 1, 9, 36, 84, 126, 126, 84, 36, 9, 1, 512, 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1

Offset: 0

Author

Clark Kimberling, Aug 05 2011

Keywords

• nonn
• tabl

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. A193787 is the mirror (obtained by reversing rows) of A193554.

Examples

			First six rows:
1
1....1
1....1....2
1....2....1....4
1....3....3....1...8
1....4....6....4...1...16
(viz., Pascal's triangle with row sum at end of each row)
		

Crossrefs

Cf. A193722, A193554.

Programs

Mathematica

z = 12; a = 1; b = 1;
p[n_, x_] := (a*x + b)^n
q[n_, x_] := 1 + x^n ; 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}]]  (* A193787 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193554 *)