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%I #10 Jul 13 2013 12:04:24
%S 1,1,1,2,2,1,3,4,3,1,4,7,7,4,1,5,11,14,11,5,1,6,16,25,25,16,6,1,7,22,
%T 41,50,41,22,7,1,8,29,63,91,91,63,29,8,1,9,37,92,154,182,154,92,37,9,
%U 1,10,46,129,246,336,336,246,129,46,10,1,11,56,175,375,582,672
%N Triangle of coefficients of polynomials u(n,x) jointly generated with A209562; see the Formula section.
%C Alternating row sums: 1,0,1,1,1,1,1,1,1,1,1,1,1,1,...
%C For a discussion and guide to related arrays, see A208510.
%H Reinhard Zumkeller, <a href="/A209561/b209561.txt">Rows n = 1..120 of triangle, flattened</a>
%F u(n,x)=x*u(n-1,x)+v(n-1,x),
%F v(n,x)=x*u(n-1,x)+v(n-1,x) +1,
%F where u(1,x)=1, v(1,x)=1.
%F T(n,n) = 1; T(n,k) = A051597(n-2,k-1), 1 <= k < n. - _Reinhard Zumkeller_, Dec 26 2012
%e First five rows:
%e 1
%e 1...1
%e 2...2...1
%e 3...4...3...1
%e 4...7...7...4...1
%e First three polynomials v(n,x): 1, 1 + x, 2 + 2x + x^2.
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t Table[Expand[u[n, x]], {n, 1, z/2}]
%t Table[Expand[v[n, x]], {n, 1, z/2}]
%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t TableForm[cu]
%t Flatten[%] (* A209561 *)
%t Table[Expand[v[n, x]], {n, 1, z}]
%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t TableForm[cv]
%t Flatten[%] (* A209562 *)
%o (Haskell)
%o a209561 n k = a209561_tabl !! (n-1) !! (k-1)
%o a209561_row n = a209561_tabl !! (n-1)
%o a209561_tabl = [1] : iterate
%o (\row -> zipWith (+) ([1] ++ row) (row ++ [0])) [1,1]
%o -- _Reinhard Zumkeller_, Dec 26 2012
%Y Cf. A209562, A208510.
%Y Cf. A083329 (row sums), A097613 (central terms).
%K nonn,tabl
%O 1,4
%A _Clark Kimberling_, Mar 10 2012
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