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A208509 Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.

%I A208509 #28 Feb 22 2022 12:43:26
%S A208509 1,3,5,1,7,5,9,14,1,11,30,7,13,55,27,1,15,91,77,9,17,140,182,44,1,19,
%T A208509 204,378,156,11,21,285,714,450,65,1,23,385,1254,1122,275,13,25,506,
%U A208509 2079,2508,935,90,1,27,650,3289,5148,2717,442,15,29,819,5005,9867
%N A208509 Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.
%F A208509 u(n,x) = u(n-1,x) + x*v(n-1,x), v(n,x) = u(n-1,x) + v(n-1,x) + 1, with u(1,x)=1, v(1,x)=1.
%F A208509 Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - _Knud Werner_, Jan 11 2022
%e A208509 First five rows:
%e A208509   1
%e A208509   3
%e A208509   5    1
%e A208509   7    5
%e A208509   9   14   1
%e A208509 First five polynomials v(n,x):
%e A208509   1
%e A208509   3
%e A208509   5 +   x
%e A208509   7 +  5x
%e A208509   9 + 14x + x^2
%t A208509 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208509 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208509 v[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A208509 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208509 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208509 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208509 TableForm[cu]
%t A208509 Flatten[%]    (* A208508 *)
%t A208509 Table[Expand[v[n, x]], {n, 1, z}]
%t A208509 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208509 TableForm[cv]
%t A208509 Flatten[%]    (* A208509 *)
%Y A208509 Columns 0 to 5: A005408, A000330, A005585, A050486, A054333, A057788.
%Y A208509 Row sums, v(n,1): A003948.
%Y A208509 Alternating row sums, v(n,-1): A090131.
%Y A208509 Cf. A208508.
%K A208509 nonn,tabf
%O A208509 1,2
%A A208509 _Clark Kimberling_, Feb 27 2012