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 A194000 Triangular array: the self-fission of (p(n,x)), where sum{F(k+1)*x^(n-k) : 0<=k<=n}, where F=A000045 (Fibonacci numbers). 3
 1, 2, 3, 3, 5, 9, 5, 8, 15, 24, 8, 13, 24, 39, 64, 13, 21, 39, 63, 104, 168, 21, 34, 63, 102, 168, 272, 441, 34, 55, 102, 165, 272, 440, 714, 1155, 55, 89, 165, 267, 440, 712, 1155, 1869, 3025, 89, 144, 267, 432, 712, 1152, 1869, 3024, 4895, 7920, 144, 233 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS See A193917 for the self-fusion of the same sequence of polynomials. (Fusion is defined at A193822; fission, at A193842; 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 A194000: 1 2....3 3....5....9 5....8....15...24 8....13...24...39...64 13...21...29...63...104...168 21...34...63...102..168...272...441 34...55...102..165..272...440...714..1155 ... col 1: A000045 col 2: A000045 col 3: A022086 col 4: A022086 col 5: A022091 col 6: A022091 right edge, d(n,n): A064831 d(n,n-1): A059840 d(n,n-2): A080097 d(n,n-3): A080143 d(n,n-4): A080144 ... Suppose n is an odd positive integer and d(n+1,x) is the polynomial matched to row n+1 of A194000 as in the Mathematica program (and definition of fission at A193842), where the first row is counted as row 0. LINKS Table of n, a(n) for n=0..56. EXAMPLE First six rows: 1 2....3 3....5....9 5....8....15...24 8....13...24...39...64 13...21...29...63...104...168 ... Referring to the matrix product for fission at A193842, the row (5,8,15,24) is the product of P(4) and QQ, where P(4)=(p(4,4), p(4,3), p(4,2), p(4,1))=(5,3,2,1); and QQ is the 4x4 matrix (1..1..2..3) (0..1..1..2) (0..0..1..1) (0..0..0..1). MATHEMATICA z = 11; p[n_, x_] := Sum[Fibonacci[k + 1]*x^(n - k), {k, 0, n}]; q[n_, x_] := p[n, x]; p1[n_, k_] := Coefficient[p[n, x], x^k]; p1[n_, 0] := p[n, x] /. x -> 0; d[n_, x_] := Sum[p1[n, k]*q[n - 1 - k, x], {k, 0, n - 1}] h[n_] := CoefficientList[d[n, x], {x}] TableForm[Table[Reverse[h[n]], {n, 0, z}]] Flatten[Table[Reverse[h[n]], {n, -1, z}]] (* A194000 *) TableForm[Table[h[n], {n, 0, z}]] Flatten[Table[h[n], {n, -1, z}]] (* A194001 *) CROSSREFS Cf. A193842, A194001, A193917, A193918. Sequence in context: A193979 A059503 A317644 * A295781 A296725 A296588 Adjacent sequences: A193997 A193998 A193999 * A194001 A194002 A194003 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Aug 11 2011 STATUS approved

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Last modified May 21 21:53 EDT 2024. Contains 372738 sequences. (Running on oeis4.)