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 A209757 Triangle of coefficients of polynomials v(n,x) jointly generated with A013609; see the Formula section. 2
 1, 3, 2, 5, 8, 4, 7, 18, 20, 8, 9, 32, 56, 48, 16, 11, 50, 120, 160, 112, 32, 13, 72, 220, 400, 432, 256, 64, 15, 98, 364, 840, 1232, 1120, 576, 128, 17, 128, 560, 1568, 2912, 3584, 2816, 1280, 256, 19, 162, 816, 2688, 6048, 9408, 9984, 6912, 2816, 512 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS For a discussion and guide to related arrays, see A208510. Subtriangle of the triangle given by (1, 2, -2, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - Philippe Deléham, Mar 24 2012 LINKS FORMULA u(n,x) = x*u(n-1,x) + x*v(n-1,x) + 1, v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x) + 1, 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 - x - 2*y*x + 2*x^2 + 2*x^2*y)/(1 - 2*x - 2*y*x + x^2 + 2*y*x^2). T(n,k) = 2*T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k) - 2*T(n-2,k-2), T(0,0) = T(1,0) = 1, T(1,1) = T(2,2) = 0, T(2,0) = 3, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k > n. T(n,k) = 2^k*binomial(n-1,k)*(2*n-k-1)/(k+1). (End) From Peter Bala, Dec 21 2014: (Start) Following remarks assume an offset of 0. T(n,k) = 2^k * A110813(n,k). Riordan array ((1+x)/(1-x)^2, 2*x/(1-x)). exp(2*x) * e.g.f. for row n = e.g.f. for diagonal n. For example, for n = 3 we have exp(2*x)*(7 + 18*x + 20*x^2/2! + 8*x^3/3!) = 7 + 32*x + 120*x^2/2! + 400*x^3/3! + 1232*x^4/4! + .... The same property holds more generally for Riordan arrays of the form (f(x), 2*x/(1-x)). (End) EXAMPLE First five rows:   1;   3,  2;   5,  8,  4;   7, 18, 20,  8;   9, 32, 56, 48, 16; First three polynomials v(n,x):   1   3 + 2x   5 + 8x + 4x^2. From Philippe Deléham, Mar 24 2012: (Start) (1, 2, -2, 1, 0, 0, ...) DELTA (0, 2, 0, 0, 0, ...) begins:   1;   1,  0;   3,  2,  0;   5,  8,  4,  0;   7, 18, 20,  8,  0;   9, 32, 56, 48, 16,  0; (End) MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1; v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1; 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[%]    (* A013609 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A209757 *) CROSSREFS Cf. A013609, A208510, A110813. Sequence in context: A132776 A249741 A246275 * A208932 A189951 A209776 Adjacent sequences:  A209754 A209755 A209756 * A209758 A209759 A209760 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 23 2012 STATUS approved

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Last modified November 28 16:34 EST 2021. Contains 349413 sequences. (Running on oeis4.)