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 A209138 Triangle of coefficients of polynomials v(n,x) jointly generated with A209137; see the Formula section. 5
 1, 1, 2, 2, 4, 3, 3, 9, 10, 5, 5, 18, 28, 22, 8, 8, 35, 68, 74, 45, 13, 13, 66, 154, 210, 177, 88, 21, 21, 122, 331, 541, 574, 397, 167, 34, 34, 222, 686, 1302, 1656, 1446, 850, 310, 55, 55, 399, 1382, 2982, 4404, 4614, 3434, 1758, 566, 89, 89, 710, 2723 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Every row begins and ends with a Fibonacci number (A000045). u(n,1) = n-th row sum = 3^(n-1). Alternating row sums: 1,-1,1,-1,1,-1,1,-1,1,-1,... For a discussion and guide to related arrays, see A208510. LINKS FORMULA u(n,x) = u(n-1,x) + (x+1)*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 11 2012: (Start) T(n,k) = A185081(n,k+1). T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k) + T(n-2,k-1) + T(n-2,k-2), T(1,0) = T(2,0) = 1, T(2,1) = 2 and T(n,k) = 0 if k < 0 or if k >= n. (End) EXAMPLE First five rows:   1;   1,  2;   2,  4,  3;   3,  9, 10,  5;   5, 18, 28, 22,  8; First three polynomials v(n,x): 1, 1 + 2x, 2 + 4x + 3x^2. From Philippe Deléham, Apr 11 2012: (Start) Triangle in A185081 begins:   1;   0,  1;   0,  1,  2;   0,  2,  4,  3;   0,  3,  9, 10,  5;   0,  5, 18, 28, 22,  8;   ... (End) MATHEMATICA u[1, x_] := 1; v[1, x_] := 1; z = 16; u[n_, x_] := u[n - 1, x] + (x + 1)*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[%]    (* A209137 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A209138 *) CROSSREFS Cf. A209137, A208510. Sequence in context: A143211 A209755 A131052 * A051297 A338756 A078317 Adjacent sequences:  A209135 A209136 A209137 * A209139 A209140 A209141 KEYWORD nonn,tabl AUTHOR Clark Kimberling, Mar 05 2012 STATUS approved

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Last modified June 18 20:45 EDT 2021. Contains 345121 sequences. (Running on oeis4.)