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

0,6

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.

Links

Table of n, a(n) for n=0..76.

Example

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)

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 *)

Crossrefs

Cf. A193722, A193554.

Sequence in context: A279186 A164799 A274451 * A072614 A287597 A238552

Adjacent sequences: A193784 A193785 A193786 * A193788 A193789 A193790

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 05 2011

Status

approved