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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 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.)