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 A209413 Triangle of coefficients of polynomials v(n,x) jointly generated with A209172; see the Formula section. 3
 1, 1, 2, 1, 3, 4, 1, 5, 7, 8, 1, 6, 17, 15, 16, 1, 8, 23, 49, 31, 32, 1, 9, 39, 72, 129, 63, 64, 1, 11, 48, 150, 201, 321, 127, 128, 1, 12, 70, 198, 501, 522, 769, 255, 256, 1, 14, 82, 338, 699, 1524, 1291, 1793, 511, 512, 1, 15, 110, 420, 1375, 2223, 4339, 3084, 4097, 1023, 1024 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS For n > 1, n-th alternating row sum = ((-1)^(n-1)*F(2n-3), where F=A000045 (Fibonacci numbers). Coefficient of x^(n-1) in u(n,x): 2^(n-1). For a discussion and guide to related arrays, see A208510. Subtriangle of the triangle T(n,k) given by (1, 0, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 11 2012 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flattened FORMULA u(n,x) = x*u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + 2x*v(n-1,x), where u(1,x)=1, v(1,x)=1. From Philippe Deléham, Mar 11 2012: (Start) As DELTA-triangle T(n,k) with 0 <= k <= n: T(n,k) = 3*T(n-1,k-1) + T(n-2,k) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(2,0) = 1, T(2,1) = 2, T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. G.f.: (1+x-3*y*x-y*x^2+2*y^2*x^2)/(1-3*y*x-(1-2y^2)*x^2). (End) EXAMPLE First five rows:   1;   1,  2;   1,  3,  4;   1,  5,  7,  8;   1,  6, 17, 15, 16; First three polynomials v(n,x):   1   1 + 2x   1 + 3x + 4x^2. From Philippe Deléham, Mar 11 2012: (Start) (1, 0, -1/2, -1/2, 0, 0, 0, ...) DELTA (0, 2, 0, 1, 0, 0, ...) begins:   1;   1,   0;   1,   2,   0;   1,   3,   4,   0;   1,   5,   7,   8,   0;   1,   6,  17,  15,  16,   0;   1,   8,  23,  49,  31,  32,   0;   1,   9,  39,  72, 129,  63,  64,   0;   1,  11,  48, 150, 201, 321, 127, 128,   0; (End) MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := x*u[n - 1, x] + v[n - 1, x]; v[n_, x_] := 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[%]    (* A209172 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A209413 *) CoefficientList[CoefficientList[Series[(1 + x - 3*y*x - y*x^2 + 2*y^2*x^2)/(1 - 3*y*x - (1 - 2 y^2)*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *) CROSSREFS Cf. A209172, A208510. Sequence in context: A224823 A078753 A119443 * A126198 A055888 A094442 Adjacent sequences:  A209410 A209411 A209412 * A209414 A209415 A209416 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 08 2012 STATUS approved

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Last modified August 11 18:37 EDT 2020. Contains 336428 sequences. (Running on oeis4.)