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Array of coefficients of polynomials v(n,x) jointly generated with A210039; see the Formula section.
3

%I #6 Mar 30 2012 18:58:16

%S 1,2,1,3,4,4,10,1,5,20,6,6,35,21,1,7,56,56,8,8,84,126,36,1,9,120,252,

%T 120,10,10,165,462,330,55,1,11,220,792,792,220,12,12,286,1287,1716,

%U 715,78,1,13,364,2002,3432,2002,364,14,14,455,3003,6435,5005,1365

%N Array of coefficients of polynomials v(n,x) jointly generated with A210039; see the Formula section.

%C Every term is a binomial coefficient.

%C Row sums: A000225

%C For a discussion and guide to related arrays, see A208510.

%F u(n,x)=u(n-1,x)+v(n-1,x)+1,

%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 Also: writing T(n,m) for the general term,

%F T(n,1)=n for n>=1;

%F T(n,k)=C(n+1,2k-1) for 1<=k<=floor[(n+2)/2].

%e First eight rows:

%e 1

%e 2...1

%e 3...4

%e 4...10...1

%e 5...20...6

%e 6...35...21....1

%e 7...56...56....8

%e 8...84...126...36...1

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

%e 1

%e 2 + x

%e 3 + 4x.

%e 4 + 10x + x^2

%e 5 + 20x + 6x^2.

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

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

%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[%] (* A210039 *)

%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[%] (* A210040 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)

%Y Cf. A210039, A208510.

%K nonn,tabf

%O 1,2

%A _Clark Kimberling_, Mar 17 2012