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 A210789 Triangle of coefficients of polynomials u(n,x) jointly generated with A210790; see the Formula section. 3
 1, 1, 1, 1, 2, 2, 1, 3, 4, 3, 1, 4, 8, 8, 5, 1, 5, 12, 18, 15, 8, 1, 6, 18, 32, 39, 28, 13, 1, 7, 24, 53, 77, 80, 51, 21, 1, 8, 32, 80, 142, 176, 160, 92, 34, 1, 9, 40, 116, 234, 352, 384, 312, 164, 55, 1, 10, 50, 160, 370, 632, 830, 812, 598, 290, 89, 1, 11, 60, 215 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS Row n starts with 1 and ends with F(n), where F=A000045 (Fibonacci numbers). Column 2: 1,2,3,4,5,6,7,8,... Row sums: A006138 Alternating row sums: signed Fibonacci numbers For a discussion and guide to related arrays, see A208510. Subtriangle of the triangle given by (1, 0, 0, -1, 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, Mar 28 2012 LINKS FORMULA u(n,x)=u(n-1,x)+x*v(n-1,x), v(n,x)=(x+2)*u(n-1,x)+(x-1)*v(n-1,x), where u(1,x)=1, v(1,x)=1. Contribution from Philippe Deléham, Mar 28 2012: (Start) As DELTA-triangle T(n,k) with 0<=k<=n: G.f.: (1+x-y*x-y*x^2-y^2*x^2)/(1-y*x-y*x^2-x^2-y^2*x^2). T(n,k) = T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + 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 five rows: 1 1...1 1...2...2 1...3...4...3 1...4...8...8...5 First three polynomials u(n,x): 1, 1 + x, 1 + 2x + 2x^2. Contribution from Philippe Deléham, Mar 28 2012: (Start) (1, 0, 0, -1, 0, 0, ...) DELTA (0, 1, 1, -1, 0, 0, ...) begins : 1 1, 0 1, 1, 0 1, 2, 2, 0 1, 3, 4, 3, 0 1, 4, 8, 8, 5, 0. (End) MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + (x + j)*v[n - 1, x] + c; d[x_] := h + x; e[x_] := p + x; v[n_, x_] := d[x]*u[n - 1, x] + e[x]*v[n - 1, x] + f; j = 0; c = 0; h = 2; p = -1; f = 0; 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[%]    (* A210789 *) cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A210790 *) Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A006138 *) Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A105476 *) Table[u[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *) Table[v[n, x] /. x -> -1, {n, 1, z}] (* [A000045] *) CROSSREFS Cf. A210790, A208510. Sequence in context: A179901 A209561 A283822 * A105809 A091594 A118032 Adjacent sequences:  A210786 A210787 A210788 * A210790 A210791 A210792 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 26 2012 STATUS approved

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Last modified January 21 22:47 EST 2020. Contains 331129 sequences. (Running on oeis4.)