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A209561 Triangle of coefficients of polynomials u(n,x) jointly generated with A209562; see the Formula section.

%I A209561 #13 Jun 24 2024 22:27:41
%S A209561 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 A209561 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 A209561 1,10,46,129,246,336,336,246,129,46,10,1,11,56,175,375,582,672
%N A209561 Triangle of coefficients of polynomials u(n,x) jointly generated with A209562; see the Formula section.
%C A209561 Alternating row sums: 1,0,1,1,1,1,1,1,1,1,1,1,1,1,...
%C A209561 For a discussion and guide to related arrays, see A208510.
%H A209561 Reinhard Zumkeller, <a href="/A209561/b209561.txt">Rows n = 1..120 of triangle, flattened</a>
%F A209561 u(n,x)=x*u(n-1,x)+v(n-1,x),
%F A209561 v(n,x)=x*u(n-1,x)+v(n-1,x) +1,
%F A209561 where u(1,x)=1, v(1,x)=1.
%F A209561 T(n,n) = 1; T(n,k) = A051597(n-2,k-1), 1 <= k < n. - _Reinhard Zumkeller_, Dec 26 2012
%e A209561 First five rows:
%e A209561 1
%e A209561 1...1
%e A209561 2...2...1
%e A209561 3...4...3...1
%e A209561 4...7...7...4...1
%e A209561 First three polynomials v(n,x): 1, 1 + x, 2 + 2x + x^2.
%t A209561 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209561 u[n_, x_] := x*u[n - 1, x] + v[n - 1, x];
%t A209561 v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A209561 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209561 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209561 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209561 TableForm[cu]
%t A209561 Flatten[%]   (* A209561 *)
%t A209561 Table[Expand[v[n, x]], {n, 1, z}]
%t A209561 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209561 TableForm[cv]
%t A209561 Flatten[%]   (* A209562 *)
%o A209561 (Haskell)
%o A209561 a209561 n k = a209561_tabl !! (n-1) !! (k-1)
%o A209561 a209561_row n = a209561_tabl !! (n-1)
%o A209561 a209561_tabl = [1] : iterate
%o A209561                (\row -> zipWith (+) ([1] ++ row) (row ++ [0])) [1,1]
%o A209561 -- _Reinhard Zumkeller_, Dec 26 2012
%Y A209561 Cf. A209562, A208510.
%Y A209561 Cf. A083329 (row sums), A097613 (central terms).
%K A209561 nonn,tabl
%O A209561 1,4
%A A209561 _Clark Kimberling_, Mar 10 2012