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Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).
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%I #17 Jan 17 2014 10:13:31

%S 1,2,3,3,5,9,5,8,15,24,8,13,24,39,64,13,21,39,63,104,168,21,34,63,102,

%T 168,272,441,34,55,102,165,272,440,714,1155,55,89,165,267,440,712,

%U 1155,1869,3025,89,144,267,432,712,1152,1869,3024,4895,7920,144,233

%N Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

%C See A193917 for the self-fusion of the same sequence of polynomials. (Fusion is defined at A193822; fission, at A193842; see A202503 and A202453 for infinite-matrix representations of fusion and fission.)

%C ...

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

%C 1

%C 1...1

%C 1...1...2

%C 1...1...2...3

%C 1...1...2...3...5

%C First eight rows of A194000:

%C 1

%C 2....3

%C 3....5....9

%C 5....8....15...24

%C 8....13...24...39...64

%C 13...21...29...63...104...168

%C 21...34...63...102..168...272...441

%C 34...55...102..165..272...440...714..1155

%C ...

%C col 1: A000045

%C col 2: A000045

%C col 3: A022086

%C col 4: A022086

%C col 5: A022091

%C col 6: A022091

%C right edge, d(n,n): A064831

%C d(n,n-1): A059840

%C d(n,n-2): A080097

%C d(n,n-3): A080143

%C d(n,n-4): A080144

%C ...

%C Suppose n is an odd positive integer and d(n+1,x) is the polynomial matched to row n+1 of A194000 as in the Mathematica program (and definition of fission at A193842), where the first row is counted as row 0.

%e First six rows:

%e 1

%e 2....3

%e 3....5....9

%e 5....8....15...24

%e 8....13...24...39...64

%e 13...21...29...63...104...168

%e ...

%e Referring to the matrix product for fission at A193842,

%e the row (5,8,15,24) is the product of P(4) and QQ, where

%e P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and

%e QQ is the 4x4 matrix

%e (1..1..2..3)

%e (0..1..1..2)

%e (0..0..1..1)

%e (0..0..0..1).

%t z = 11;

%t p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];

%t q[n_, x_] := p[n, x];

%t p1[n_, k_] := Coefficient[p[n, x], x^k];

%t p1[n_, 0] := p[n, x] /. x -> 0;

%t d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]

%t h[n_] := CoefficientList[d[n, x], {x}]

%t TableForm[Table[Reverse[h[n]], {n, 0, z}]]

%t Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A194000 *)

%t TableForm[Table[h[n], {n, 0, z}]]

%t Flatten[Table[h[n], {n, -1, z}]] (* A194001 *)

%Y Cf. A193842, A194001, A193917, A193918.

%K nonn,tabl

%O 0,2

%A _Clark Kimberling_, Aug 11 2011