A193904 - OEIS

%I #7 Mar 30 2012 18:57:39

%S 1,2,1,8,6,3,32,24,14,7,128,96,56,30,15,512,384,224,120,62,31,2048,

%T 1536,896,480,248,126,63,8192,6144,3584,1920,992,504,254,127,32768,

%U 24576,14336,7680,3968,2016,1016,510,255,131072,98304,57344,30720,15872

%N 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.

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

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

%C 1

%C 1...2

%C 1...2...4

%C 1...2...4...8

%C 1...2...4...8...16

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

%C 1

%C 2...1

%C 4...2...1

%C 8...4...2...1

%C 16..8...4...2..1

%e First six rows of A193904:

%e 1

%e 2....1

%e 8....6....3

%e 32...24...14...7

%e 128..96...56...30...15

%e 512..384..224..120..62..31

%t z = 12;

%t p[n_, x_] := x*p[n - 1, x] + 2^n; p[0, x_] := 1;

%t q[n_, x_] := 2 x*q[n - 1, x] + 1; q[0, x_] := 1;

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193904 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]]  (* A193905 *)

%Y Cf. A193722, A193905.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 08 2011