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 A210868 Triangle of coefficients of polynomials u(n,x) jointly generated with A210869; see the Formula section. 2
 1, 1, 1, 1, 1, 2, 1, 2, 2, 3, 1, 2, 5, 3, 5, 1, 3, 5, 10, 5, 8, 1, 3, 9, 10, 20, 8, 13, 1, 4, 9, 22, 20, 38, 13, 21, 1, 4, 14, 22, 51, 38, 71, 21, 34, 1, 5, 14, 40, 51, 111, 71, 130, 34, 55, 1, 5, 20, 40, 105, 111, 233, 130, 235, 55, 89, 1, 6, 20, 65, 105, 256, 233, 474 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS In row n the first two terms are 1 and floor(n/2), and the last two, for n>1, are F(n-1) and F(n), where F = A000045 (Fibonacci numbers). Row sums: 1,2,4,8,16,32,...; A000079. Alternating row sums:  A151575 For a discussion and guide to related arrays, see A208510. Subtriangle of the triangle given by (1, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 02 2012 LINKS FORMULA u(n,x)=u(n-1,x)+x*v(n-1,x), v(n,x)=(x+1)*u(n-1,x)+(x-1)*v(n-1,x), where u(1,x)=1, v(1,x)=1. From Philippe Deléham, Apr 02 2012: (Start) As DELTA-triangle T(n,k) with 0<=k<=n : G.f.: (1+x-y*x-y^2*x^2)/(1-y*x-y^2*x^2-x^2). T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End) EXAMPLE First six rows:   1   1...1   1...1...2   1...2...2...3   1...2...5...3....5   1...3...5...10...5...8 First three polynomials u(n,x): 1, 1 + x, 1 + x + 2x^2. (1, 0, -1, 0, 0, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, 0, ...) begins : 1 1, 0 1, 1, 0 1, 1, 2, 0 1, 2, 2, 3, 0 1, 2, 5, 3, 5, 0 1, 3, 5, 10, 5, 8, 0. - Philippe Deléham, Apr 02 2012 MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 14; u[n_, x_] := u[n - 1, x] + x*v[n - 1, x]; v[n_, x_] := (x + n)*u[n - 1, x] + x*v[n - 1, x] - x; Table[Expand[u[n, x]], {n, 1, z/2}] Table[Expand[v[n, x]], {n, 1, z/2}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%]   (* A210866 *) cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]   (* A210867 *) CROSSREFS Cf. A210867, A208510. Sequence in context: A290399 A109400 A202389 * A176853 A261787 A302480 Adjacent sequences:  A210865 A210866 A210867 * A210869 A210870 A210871 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 29 2012 STATUS approved

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Last modified October 14 06:51 EDT 2019. Contains 327995 sequences. (Running on oeis4.)