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A193904
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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.
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2
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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EXAMPLE
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1
2....1
8....6....3
32...24...14...7
128..96...56...30...15
512..384..224..120..62..31
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MATHEMATICA
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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 *)
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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