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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).
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%I #16 Mar 30 2012 18:57:39

%S 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,

%T 168,13,21,39,63,104,168,273,441,21,34,63,102,168,272,441,714,1155,34,

%U 55,102,165,272,440,714,1155,1870,3025,55,89,165,267,440,712,1155

%N 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).

%C 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.)

%C First five rows of P (triangle of coefficients of polynomials p(n,x)):

%C 1

%C 1...1

%C 1...1...2

%C 1...1...2...3

%C 1...1...2...3...5

%C First eight rows of A193917:

%C 1

%C 1...1

%C 1...2...3

%C 2...3...6...9

%C 3...5...9...15...24

%C 5...8...15..24...40...64

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

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

%C ...

%C col 1: A000045

%C col 2: A000045

%C col 3: A022086

%C col 4: A022086

%C col 5: A022091

%C col 6: A022091

%C col 7: A022355

%C col 8: A022355

%C right edge, w(n,n): A064831

%C w(n,n-1): A001654

%C w(n,n-2): A064831

%C w(n,n-3): A059840

%C w(n,n-4): A080097

%C w(n,n-5): A080143

%C w(n,n-6): A080144

%C 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.

%e First six rows:

%e 1

%e 1...1

%e 1...2...3

%e 2...3...6....9

%e 3...5...9....15...24

%e 5...8...15...24...40...64

%t z = 12;

%t p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];

%t q[n_, x_] := p[n, x];

%t t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;

%t w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1

%t g[n_] := CoefficientList[w[n, x], {x}]

%t TableForm[Table[Reverse[g[n]], {n, -1, z}]]

%t Flatten[Table[Reverse[g[n]], {n, -1, z}]] (* A193917 *)

%t TableForm[Table[g[n], {n, -1, z}]]

%t Flatten[Table[g[n], {n, -1, z}]] (* A193918 *)

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

%K nonn,tabl

%O 0,5

%A _Clark Kimberling_, Aug 09 2011