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 A209415 Triangle of coefficients of polynomials u(n,x) jointly generated with A209416; see the Formula section. 6
 1, 1, 1, 1, 3, 1, 1, 4, 6, 1, 1, 6, 11, 10, 1, 1, 7, 21, 25, 15, 1, 1, 9, 30, 57, 50, 21, 1, 1, 10, 45, 99, 133, 91, 28, 1, 1, 12, 58, 168, 275, 280, 154, 36, 1, 1, 13, 78, 250, 523, 675, 546, 246, 45, 1, 1, 15, 95, 370, 885, 1433, 1509, 1002, 375, 55, 1, 1, 16, 120, 505, 1435, 2718, 3564, 3135, 1749, 550, 66, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,5 COMMENTS For a discussion and guide to related arrays, see A208510. Subtriangle of the triangle given by (1, 0, 1, -2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Apr 02 2012 Up to reflection at the vertical axis, the triangle of numbers given here coincides with the triangle given in A208334, i.e., the numbers are the same just read row-wise in the opposite direction. - Christine Bessenrodt, Jul 21 2012 LINKS G. C. Greubel, Table of n, a(n) for the first 50 rows, flatened FORMULA u(n,x) = x*u(n-1,x) + v(n-1,x), v(n,x) = (x+1)*u(n-1,x) + x*v(n-1,x), where u(1,x)=1, v(1,x)=1. From Philippe Deléham, Apr 02 2012: (Start) As DELTA-triangle T(n,k) with 0 <= k <= n: G.f.: (1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2). T(n,k) = 2*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,  3,  1;   1,  4,  6,  1;   1,  6, 11, 10,  1; First three polynomials v(n,x): 1, 1 + x, 1 + 3x + x^2. From Philippe Deléham, Apr 02 2012: (Start) (1, 0, 1, -2, 0, 0, 0, ...) DELTA (0, 1, 0, 1, 0, 0, 0, ...) begins:   1;   1,   0;   1,   1,   0;   1,   3,   1,   0;   1,   4,   6,   1,   0;   1,   6,  11,  10,   1,   0;   1,   7,  21,  25,  15,   1,   0;   1,   9,  30,  57,  50,  21,   1,   0;   1,  10,  45,  99, 133,  91,  28,   1,   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_] := (x + 1)*u[n - 1, x] + 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[%]    (* A209415 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A209416 *) CoefficientList[CoefficientList[Series[(1 + x - 2*y*x - 2*y*x^2 + y^2*x^2)/(1 - 2*y*x - x^2 - y*x^2 + y^2*x^2), {x, 0, 10}, {y, 0, 10}], x], y] // Flatten (* G. C. Greubel, Jan 03 2018 *) CROSSREFS Cf. A209416, A208510, A208334. Sequence in context: A133380 A343168 A105687 * A058879 A208344 A209172 Adjacent sequences:  A209412 A209413 A209414 * A209416 A209417 A209418 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 09 2012 STATUS approved

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Last modified December 4 02:40 EST 2021. Contains 349469 sequences. (Running on oeis4.)