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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). 3
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 (list; table; graph; refs; listen; history; text; internal format)
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: A328841 A276008 A331171 * A089135 A215412 A227585

Adjacent sequences:  A193914 A193915 A193916 * A193918 A193919 A193920

KEYWORD

nonn,tabl

AUTHOR

Clark Kimberling, Aug 09 2011

STATUS

approved

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Last modified June 2 08:00 EDT 2020. Contains 334767 sequences. (Running on oeis4.)