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 A208760 Triangle of coefficients of polynomials v(n,x) jointly generated with A208759; see the Formula section. 3
 1, 1, 3, 1, 5, 8, 1, 7, 20, 22, 1, 9, 36, 72, 60, 1, 11, 56, 158, 244, 164, 1, 13, 80, 288, 632, 796, 448, 1, 15, 108, 470, 1320, 2376, 2528, 1224, 1, 17, 140, 712, 2420, 5592, 8544, 7872, 3344, 1, 19, 176, 1022, 4060, 11372, 22368, 29712, 24144, 9136 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For a discussion and guide to related arrays, see A208510. Subtriangle of the triangle given by (1, 0, -1/3, 1/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 18 2012 LINKS Table of n, a(n) for n=1..55. FORMULA u(n,x) = u(n-1,x) + 2x*v(n-1,x), v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x), where u(1,x)=1, v(1,x)=1. From Philippe Deléham, Mar 18 2012: (Start) As DELTA-triangle with 0 <= k <= n: G.f.: (1-2*y*x+y*x^2-2*y^2*x^2)/(1-x-2*y*x-2*y^2*x^2). T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + 2*T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 3 and T(n,k) = 0 if k < 0 or if k > n. (End) EXAMPLE First five rows: 1; 1, 3; 1, 5, 8; 1, 7, 20, 22; 1, 9, 36, 72, 60; First five polynomials v(n,x): 1 1 + 3x 1 + 5x + 8x^2 1 + 7x + 20x^2 + 22x^3 1 + 9x + 36x^2 + 72x^3 + 60x^4 From Philippe Deléham, Mar 18 2012: (Start) (1, 0, -1/3, 1/3, 0, 0, ...) DELTA (0, 3, -1/3, -2/3, 0, 0, ...) begins: 1; 1, 0; 1, 3, 0; 1, 5, 8, 0; 1, 7, 20, 22, 0; 1, 9, 36, 72, 60, 0; 1, 11, 56, 158, 244, 164, 0; (End) 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_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, 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[%] (* A208759 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A208760 *) CROSSREFS Cf. A208759, A208510. Sequence in context: A261712 A038738 A210741 * A340156 A340242 A116647 Adjacent sequences: A208757 A208758 A208759 * A208761 A208762 A208763 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 02 2012 STATUS approved

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Last modified August 4 17:39 EDT 2024. Contains 374923 sequences. (Running on oeis4.)