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 A207536 Triangle of coefficients of polynomials u(n,x) jointly generated with A105070; see Formula section. 3
 1, 1, 2, 1, 6, 1, 12, 4, 1, 20, 20, 1, 30, 60, 8, 1, 42, 140, 56, 1, 56, 280, 224, 16, 1, 72, 504, 672, 144, 1, 90, 840, 1680, 720, 32, 1, 110, 1320, 3696, 2640, 352, 1, 132, 1980, 7392, 7920, 2112, 64, 1, 156, 2860, 13728, 20592, 9152, 832, 1, 182, 4004 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Subtriangle of the triangle given by (1, 0, 1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 08 2012 LINKS FORMULA u(n,x)=u(n-1,x)+2x*v(n-1,x) and v(n,x)=u(n-1,x)+v(n-1,x), where u(1,x)=1, v(1,x)=1. Contribution from Philippe Deléham, Apr 08 2012 . (Start) As DELTA-triangle T(n,k) with 0<=k<=n : G.f.: (1-x)/(1-2*x+x^2-2*y*x^2) . T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n. T(n,k) = A034839(n,k)*2^k = binomial(n,2*k)*2^k . (End) EXAMPLE First seven rows: 1... 1...2 1...6 1...12...4 1...20...20 1...30...60...8 1...42...140..56 (1, 0, 1, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, ...) begins : 1 1, 0 1, 2, 0 1, 6, 0, 0 1, 12, 4, 0, 0 1, 20, 20, 0, 0, 0 1, 30, 60, 8, 0, 0, 0 1, 42, 140, 56, 0, 0, 0, 0 . - Philippe Deléham, Apr 08 2012 MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x] v[n_, x_] := u[n - 1, x] + v[n - 1, x] Table[Factor[u[n, x]], {n, 1, z}] Table[Factor[v[n, x]], {n, 1, z}] cu = Table[CoefficientList[u[n, x], x], {n, 1, z}]; TableForm[cu] Flatten[%]  (* A207536 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]  (* A105070 *) CROSSREFS Cf. A105070. Sequence in context: A139625 A053785 A233809 * A060173 A059344 A109193 Adjacent sequences:  A207533 A207534 A207535 * A207537 A207538 A207539 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Feb 18 2012 STATUS approved

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Last modified December 14 07:15 EST 2018. Contains 318090 sequences. (Running on oeis4.)