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 A209696 Triangle of coefficients of polynomials v(n,x) jointly generated with A209695; see the Formula section. 4
 1, 1, 3, 1, 6, 7, 1, 9, 23, 17, 1, 12, 48, 76, 41, 1, 15, 82, 204, 233, 99, 1, 18, 125, 428, 765, 682, 239, 1, 21, 177, 775, 1907, 2649, 1935, 577, 1, 24, 238, 1272, 4010, 7656, 8680, 5368, 1393, 1, 27, 308, 1946, 7506, 18358, 28548, 27312, 14641, 3363 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Alternating row sums: 1,-2,2,-2,2,-2,2,-2,... For a discussion and guide to related arrays, see A208510. Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012 Mirror image of triangle in A054458. - Philippe Deléham, Mar 24 2012 LINKS Table of n, a(n) for n=1..55. FORMULA u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x), v(n,x) = 2x*u(n-1,x) + (x+1)*v(n-1,x), where u(1,x)=1, v(1,x)=1. From Philippe Deléham, Mar 24 2012: (Start) As DELTA-triangle T(n,k) with 0 <= k <= n: G.f.: (1-2*y*x-y^2*x^2)/(1-x-2*y*x-y*x^2-y^2*x^2). T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) + 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, 6, 7; 1, 9, 23, 17; 1, 12, 48, 76, 41; First three polynomials v(n,x): 1 1 + 3x 1 + 6x + 7x^2. From Philippe Deléham, Mar 24 2012: (Start) (1, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, ...) begins: 1; 1, 0; 1, 3, 0; 1, 6, 7, 0; 1, 9, 23, 17, 0; 1, 12, 48, 76, 41, 0; (End) MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x]; v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*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[%] (* A209695 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%] (* A209696 *) CROSSREFS Cf. A054458, A209695, A208510. Sequence in context: A124929 A208766 A259454 * A338995 A359574 A210749 Adjacent sequences: A209693 A209694 A209695 * A209697 A209698 A209699 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 13 2012 STATUS approved

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Last modified June 20 07:26 EDT 2024. Contains 373512 sequences. (Running on oeis4.)