A193904 Triangular array: the fusion of (p(n,x)) by (q(n,x)), where p(n,x)=x*p(n-1,x)+2^n with p(0,x)=1, and q(n,x)=2x*q(n-1,x)+1 with q(0,x)=1.

1, 2, 1, 8, 6, 3, 32, 24, 14, 7, 128, 96, 56, 30, 15, 512, 384, 224, 120, 62, 31, 2048, 1536, 896, 480, 248, 126, 63, 8192, 6144, 3584, 1920, 992, 504, 254, 127, 32768, 24576, 14336, 7680, 3968, 2016, 1016, 510, 255, 131072, 98304, 57344, 30720, 15872

Offset

0,2

Comments

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

First five rows of P, from coefficients of p(n,x):

1

1...2

1...2...4

1...2...4...8

1...2...4...8...16

First five rows of Q, from coefficients of q(n,x):

1

2...1

4...2...1

8...4...2...1

16..8...4...2..1

Links

Table of n, a(n) for n=0..49.

Example

First six rows of A193904:
1
2....1
8....6....3
32...24...14...7
128..96...56...30...15
512..384..224..120..62..31

Mathematica

z = 12;
p[n_, x_] := x*p[n - 1, x] + 2^n; p[0, x_] := 1;
q[n_, x_] := 2 x*q[n - 1, x] + 1; q[0, x_] := 1;
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}]]  (* A193904 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193905 *)

Crossrefs

Cf. A193722, A193905.

Sequence in context: A272983 A195698 A193908 * A154897 A198577 A011135

Adjacent sequences: A193901 A193902 A193903 * A193905 A193906 A193907

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 08 2011

Status

approved