A193794 Triangular array: the fusion of polynomial sequences P and Q given by p(n,x)=(3x+1)^n and q(n,x)=1+x^n.

1, 1, 1, 1, 3, 4, 1, 6, 9, 16, 1, 9, 27, 27, 64, 1, 12, 54, 108, 81, 256, 1, 15, 90, 270, 405, 243, 1024, 1, 18, 135, 540, 1215, 1458, 729, 4096, 1, 21, 189, 945, 2835, 5103, 5103, 2187, 16384, 1, 24, 252, 1512, 5670, 13608, 20412, 17496, 6561, 65536, 1

Offset

0,5

Comments

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

Links

Table of n, a(n) for n=0..55.

Example

First six rows:
1
1...1
1...3....4
1...6....9....16
1...9....27...27....64
1...12...54...108...81...256

Mathematica

z = 9; a = 3; 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}]]  (* A193794 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193795 *)

Crossrefs

Cf. A193722, A193795.

Sequence in context: A359268 A255604 A132700 * A281862 A366795 A092261

Adjacent sequences: A193791 A193792 A193793 * A193795 A193796 A193797

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 05 2011

Status

approved