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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).

%I A193917 #16 Mar 30 2012 18:57:39
%S 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,
%T A193917 168,13,21,39,63,104,168,273,441,21,34,63,102,168,272,441,714,1155,34,
%U A193917 55,102,165,272,440,714,1155,1870,3025,55,89,165,267,440,712,1155
%N 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).
%C A193917 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 A193917 First five rows of P (triangle of coefficients of polynomials p(n,x)):
%C A193917 1
%C A193917 1...1
%C A193917 1...1...2
%C A193917 1...1...2...3
%C A193917 1...1...2...3...5
%C A193917 First eight rows of A193917:
%C A193917 1
%C A193917 1...1
%C A193917 1...2...3
%C A193917 2...3...6...9
%C A193917 3...5...9...15...24
%C A193917 5...8...15..24...40...64
%C A193917 8...13..24..39...64...104..168
%C A193917 13..21..39..63...104..168..273..441
%C A193917 ...
%C A193917 col 1:  A000045
%C A193917 col 2:  A000045
%C A193917 col 3:  A022086
%C A193917 col 4:  A022086
%C A193917 col 5:  A022091
%C A193917 col 6:  A022091
%C A193917 col 7:  A022355
%C A193917 col 8:  A022355
%C A193917 right edge, w(n,n):  A064831
%C A193917 w(n,n-1):  A001654
%C A193917 w(n,n-2):  A064831
%C A193917 w(n,n-3):  A059840
%C A193917 w(n,n-4):  A080097
%C A193917 w(n,n-5):  A080143
%C A193917 w(n,n-6):  A080144
%C A193917 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 A193917 First six rows:
%e A193917 1
%e A193917 1...1
%e A193917 1...2...3
%e A193917 2...3...6....9
%e A193917 3...5...9....15...24
%e A193917 5...8...15...24...40...64
%t A193917 z = 12;
%t A193917 p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}];
%t A193917 q[n_, x_] := p[n, x];
%t A193917 t[n_, k_] := Coefficient[p[n, x], x^k]; t[n_, 0] := p[n, x] /. x -> 0;
%t A193917 w[n_, x_] := Sum[t[n, k]*q[n + 1 - k, x], {k, 0, n}]; w[-1, x_] := 1
%t A193917 g[n_] := CoefficientList[w[n, x], {x}]
%t A193917 TableForm[Table[Reverse[g[n]], {n, -1, z}]]
%t A193917 Flatten[Table[Reverse[g[n]], {n, -1, z}]]  (* A193917 *)
%t A193917 TableForm[Table[g[n], {n, -1, z}]]
%t A193917 Flatten[Table[g[n], {n, -1, z}]]  (* A193918 *)
%Y A193917 Cf. A193722, A064831, A193918, A194000, A194001.
%K A193917 nonn,tabl
%O A193917 0,5
%A A193917 _Clark Kimberling_, Aug 09 2011