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Triangle of coefficients of polynomials v(n,x) jointly generated with A208508; see the Formula section.
4

%I #28 Feb 22 2022 12:43:26

%S 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 204,378,156,11,21,285,714,450,65,1,23,385,1254,1122,275,13,25,506,

%U 2079,2508,935,90,1,27,650,3289,5148,2717,442,15,29,819,5005,9867

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

%F 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 Conjecture: T(n,k) = binomial(n-1,2*k+1) + binomial(n,2*k+1). - _Knud Werner_, Jan 11 2022

%e First five rows:

%e 1

%e 3

%e 5 1

%e 7 5

%e 9 14 1

%e First five polynomials v(n,x):

%e 1

%e 3

%e 5 + x

%e 7 + 5x

%e 9 + 14x + x^2

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

%t v[n_, 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[%] (* A208508 *)

%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[%] (* A208509 *)

%Y Columns 0 to 5: A005408, A000330, A005585, A050486, A054333, A057788.

%Y Row sums, v(n,1): A003948.

%Y Alternating row sums, v(n,-1): A090131.

%Y Cf. A208508.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Feb 27 2012