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A209421 Triangle of coefficients of polynomials u(n,x) jointly generated with A209422; see the Formula section.

%I A209421 #16 Jan 04 2018 17:31:41
%S A209421 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,
%T A209421 44,30,13,8,1,1,67,97,91,63,39,15,9,1,1,109,176,179,140,85,49,17,10,1,
%U A209421 1,177,315,349,291,201,110,60,19,11,1,1,287,559,667,601,437,275,138,72,21,12,1,1
%N A209421 Triangle of coefficients of polynomials u(n,x) jointly generated with A209422; see the Formula section.
%C A209421 For a discussion and guide to related arrays, see A208510.
%H A209421 G. C. Greubel, <a href="/A209421/b209421.txt">Table of n, a(n) for the first 100 rows, flattened</a>
%F A209421 u(n,x) = x*u(n-1,x) + v(n-1,x),
%F A209421 v(n,x) = u(n-1,x) + v(n-1,x) +1,
%F A209421 where u(1,x) = 1, v(1,x) = 1.
%F A209421 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
%F A209421 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
%e A209421 First five rows:
%e A209421   1
%e A209421   1 1
%e A209421   3 1 1
%e A209421   5 4 1 1
%e A209421   9 7 5 1 1
%e A209421 First three polynomials v(n,x): 1, 1 + x, 3 + x + x^2.
%t A209421 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209421 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209421 v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A209421 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209421 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209421 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209421 TableForm[cu]
%t A209421 Flatten[%]    (* A209421 *)
%t A209421 Table[Expand[v[n, x]], {n, 1, z}]
%t A209421 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209421 TableForm[cv]
%t A209421 Flatten[%]    (* A209422 *)
%t A209421 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 *)
%Y A209421 Cf. A001595 (column 1), A209422, A208510, A212804.
%K A209421 nonn,tabl,easy
%O A209421 1,4
%A A209421 _Clark Kimberling_, Mar 09 2012