A193917 Triangular array: the self-fusion of (p(n,x)), where p(n,x)=sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers).

1, 1, 1, 1, 2, 3, 2, 3, 6, 9, 3, 5, 9, 15, 24, 5, 8, 15, 24, 40, 64, 8, 13, 24, 39, 64, 104, 168, 13, 21, 39, 63, 104, 168, 273, 441, 21, 34, 63, 102, 168, 272, 441, 714, 1155, 34, 55, 102, 165, 272, 440, 714, 1155, 1870, 3025, 55, 89, 165, 267, 440, 712, 1155

Offset

0,5

Comments

See A193722 for the definition of fusion of two sequences of polynomials or triangular arrays. (Fusion is defined at A193822; fission, at A193742; 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 A193917:

1

1...1

1...2...3

2...3...6...9

3...5...9...15...24

5...8...15..24...40...64

8...13..24..39...64...104..168

13..21..39..63...104..168..273..441

...

col 1: A000045

col 2: A000045

col 3: A022086

col 4: A022086

col 5: A022091

col 6: A022091

col 7: A022355

col 8: A022355

right edge, w(n,n): A064831

w(n,n-1): A001654

w(n,n-2): A064831

w(n,n-3): A059840

w(n,n-4): A080097

w(n,n-5): A080143

w(n,n-6): A080144

Suppose n is an even positive integer and w(n+1,x) is the polynomial matched to row n+1 of A193917 as in the Mathematica program (and definition of fusion at A193722), where the first row is counted as row 0.

Links

Table of n, a(n) for n=0..61.

Example

First six rows:
1
1...1
1...2...3
2...3...6....9
3...5...9....15...24
5...8...15...24...40...64

Mathematica

z = 12;
p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
q[n_, x_] := p[n, x];
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}]]  (* A193917 *)
TableForm[Table[g[n], {n, -1, z}]]
Flatten[Table[g[n], {n, -1, z}]]  (* A193918 *)

Crossrefs

Cf. A193722, A064831, A193918, A194000, A194001.

Sequence in context: A276008 A331171 A336953 * A089135 A215412 A227585

Adjacent sequences: A193914 A193915 A193916 * A193918 A193919 A193920

Keyword

nonn,tabl

Author

Clark Kimberling, Aug 09 2011

Status

approved