

A194000


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


3



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, 168, 272, 441, 34, 55, 102, 165, 272, 440, 714, 1155, 55, 89, 165, 267, 440, 712, 1155, 1869, 3025, 89, 144, 267, 432, 712, 1152, 1869, 3024, 4895, 7920, 144, 233
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OFFSET

0,2


COMMENTS

See A193917 for the selffusion of the same sequence of polynomials. (Fusion is defined at A193822; fission, at A193842; see A202503 and A202453 for infinitematrix representations of fusion and fission.)
...
First five rows of P (triangle of coefficients of polynomials p(n,x)):
1
1...1
1...1...2
1...1...2...3
1...1...2...3...5
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
21...34...63...102..168...272...441
34...55...102..165..272...440...714..1155
...
...
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.


LINKS



EXAMPLE

First six rows:
1
2....3
3....5....9
5....8....15...24
8....13...24...39...64
13...21...29...63...104...168
...
Referring to the matrix product for fission at A193842,
the row (5,8,15,24) is the product of P(4) and QQ, where
P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and
QQ is the 4x4 matrix
(1..1..2..3)
(0..1..1..2)
(0..0..1..1)
(0..0..0..1).


MATHEMATICA

z = 11;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n  k), {k, 0, n}];
q[n_, x_] := p[n, x];
p1[n_, k_] := Coefficient[p[n, x], x^k];
p1[n_, 0] := p[n, x] /. x > 0;
d[n_, x_] := Sum[p1[n, k]*q[n  1  k, x], {k, 0, n  1}]
h[n_] := CoefficientList[d[n, x], {x}]
TableForm[Table[Reverse[h[n]], {n, 0, z}]]
Flatten[Table[Reverse[h[n]], {n, 1, z}]] (* A194000 *)
TableForm[Table[h[n], {n, 0, z}]]
Flatten[Table[h[n], {n, 1, z}]] (* A194001 *)


CROSSREFS



KEYWORD



AUTHOR



STATUS

approved



