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 A209421 Triangle of coefficients of polynomials u(n,x) jointly generated with A209422; see the Formula section. 3
 1, 1, 1, 3, 1, 1, 5, 4, 1, 1, 9, 7, 5, 1, 1, 15, 15, 9, 6, 1, 1, 25, 28, 22, 11, 7, 1, 1, 41, 53, 44, 30, 13, 8, 1, 1, 67, 97, 91, 63, 39, 15, 9, 1, 1, 109, 176, 179, 140, 85, 49, 17, 10, 1, 1, 177, 315, 349, 291, 201, 110, 60, 19, 11, 1, 1, 287, 559, 667, 601, 437, 275, 138, 72, 21, 12, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS For a discussion and guide to related arrays, see A208510. LINKS G. C. Greubel, Table of n, a(n) for the first 100 rows, flattened FORMULA u(n,x) = x*u(n-1,x) + v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x) +1, where u(1,x) = 1, v(1,x) = 1. Riordan array (f(z), z*g(z)) where f(z) = (1 - z + z^2)/(1 - 2*z + z^3) is the o.g.f. for A001595 and g(z) = (1 - z)/(1 - z - z^2) is the o.g.f. for A212804, a variant of the Fibonacci numbers. - Peter Bala, Dec 30 2015 G.f.: (1 + (1 - x)*t - t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)) = 1 + (1+x)*t + (3+x+x^2)*t^2 + ... . - G. C. Greubel, Jan 03 2018 EXAMPLE First five rows:   1   1 1   3 1 1   5 4 1 1   9 7 5 1 1 First three polynomials v(n,x): 1, 1 + x, 3 + x + x^2. 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] + 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[%]    (* A209421 *) Table[Expand[v[n, x]], {n, 1, z}] cv = Table[CoefficientList[v[n, x], x], {n, 1, z}]; TableForm[cv] Flatten[%]    (* A209422 *) CoefficientList[CoefficientList[Series[(1 - t + t^2)/((1 - t)*(1 - (x + 1)*t + (x - 1)*t^2)), {t, 0, 10}], t], x] // Flatten (* G. C. Greubel, Jan 03 2018 *) CROSSREFS Cf. A001595 (column 1), A209422, A208510, A212804. Sequence in context: A208510 A131767 A026780 * A320435 A275421 A243576 Adjacent sequences:  A209418 A209419 A209420 * A209422 A209423 A209424 KEYWORD nonn,tabl,easy AUTHOR Clark Kimberling, Mar 09 2012 STATUS approved

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Last modified February 19 19:06 EST 2020. Contains 332047 sequences. (Running on oeis4.)