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).

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

Offset

0,2

Comments

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.)

...

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

First eight rows of A194000:

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

...

col 1: A000045

col 2: A000045

col 3: A022086

col 4: A022086

col 5: A022091

col 6: A022091

right edge, d(n,n): A064831

d(n,n-1): A059840

d(n,n-2): A080097

d(n,n-3): A080143

d(n,n-4): A080144

...

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

Table of n, a(n) for n=0..56.

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

Cf. A193842, A194001, A193917, A193918.

Sequence in context: A193979 A059503 A317644 * A295781 A296725 A296588

Adjacent sequences: A193997 A193998 A193999 * A194001 A194002 A194003

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 11 2011

Status

approved