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A194000 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).

%I A194000 #17 Jan 17 2014 10:13:31
%S A194000 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 A194000 168,272,441,34,55,102,165,272,440,714,1155,55,89,165,267,440,712,
%U A194000 1155,1869,3025,89,144,267,432,712,1152,1869,3024,4895,7920,144,233
%N A194000 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 A194000 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 A194000 ...
%C A194000 First five rows of P (triangle of coefficients of polynomials p(n,x)):
%C A194000 1
%C A194000 1...1
%C A194000 1...1...2
%C A194000 1...1...2...3
%C A194000 1...1...2...3...5
%C A194000 First eight rows of A194000:
%C A194000 1
%C A194000 2....3
%C A194000 3....5....9
%C A194000 5....8....15...24
%C A194000 8....13...24...39...64
%C A194000 13...21...29...63...104...168
%C A194000 21...34...63...102..168...272...441
%C A194000 34...55...102..165..272...440...714..1155
%C A194000 ...
%C A194000 col 1:  A000045
%C A194000 col 2:  A000045
%C A194000 col 3:  A022086
%C A194000 col 4:  A022086
%C A194000 col 5:  A022091
%C A194000 col 6:  A022091
%C A194000 right edge, d(n,n):  A064831
%C A194000 d(n,n-1):  A059840
%C A194000 d(n,n-2):  A080097
%C A194000 d(n,n-3):  A080143
%C A194000 d(n,n-4):  A080144
%C A194000 ...
%C A194000 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 A194000 First six rows:
%e A194000 1
%e A194000 2....3
%e A194000 3....5....9
%e A194000 5....8....15...24
%e A194000 8....13...24...39...64
%e A194000 13...21...29...63...104...168
%e A194000 ...
%e A194000 Referring to the matrix product for fission at A193842,
%e A194000 the row (5,8,15,24) is the product of P(4) and QQ, where
%e A194000 P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and
%e A194000 QQ is the 4x4 matrix
%e A194000 (1..1..2..3)
%e A194000 (0..1..1..2)
%e A194000 (0..0..1..1)
%e A194000 (0..0..0..1).
%t A194000 z = 11;
%t A194000 p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A194000 q[n_, x_] := p[n, x];
%t A194000 p1[n_, k_] := Coefficient[p[n, x], x^k];
%t A194000 p1[n_, 0] := p[n, x] /. x -> 0;
%t A194000 d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}]
%t A194000 h[n_] := CoefficientList[d[n, x], {x}]
%t A194000 TableForm[Table[Reverse[h[n]], {n, 0, z}]]
%t A194000 Flatten[Table[Reverse[h[n]], {n, -1, z}]]  (* A194000 *)
%t A194000 TableForm[Table[h[n], {n, 0, z}]]
%t A194000 Flatten[Table[h[n], {n, -1, z}]]  (* A194001 *)
%Y A194000 Cf. A193842, A194001, A193917, A193918.
%K A194000 nonn,tabl
%O A194000 0,2
%A A194000 _Clark Kimberling_, Aug 11 2011