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%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
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