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A210039 Array of coefficients of polynomials u(n,x) jointly generated with A210040; see the Formula section.

%I A210039 #8 Sep 02 2018 09:47:39
%S A210039 1,3,6,1,10,5,15,15,1,21,35,7,28,70,28,1,36,126,84,9,45,210,210,45,1,
%T A210039 55,330,462,165,11,66,495,924,495,66,1,78,715,1716,1287,286,13,91,
%U A210039 1001,3003,3003,1001,91,1,105,1365,5005,6435,3003,455,15,120,1820
%N A210039 Array of coefficients of polynomials u(n,x) jointly generated with A210040; see the Formula section.
%C A210039 Every term is a binomial coefficient.
%C A210039 Row sums:  A000225
%C A210039 For a discussion and guide to related arrays, see A208510.
%F A210039 u(n,x)=u(n-1,x)+v(n-1,x)+1,
%F A210039 v(n,x)=x*u(n-1,x)+v(n-1,x)+1,
%F A210039 where u(1,x)=1, v(1,x)=1.
%F A210039 Also, writing the general term as T(n,m),
%F A210039 T(n,k)=C(n,2k) for 1<=k<=floor[(n+1)/2], for n>=1.
%e A210039 First eight rows:
%e A210039 1
%e A210039 3
%e A210039 6....1
%e A210039 10...5
%e A210039 15...15....1
%e A210039 21...35....7
%e A210039 28...70....28...1
%e A210039 36...126...84...9
%e A210039 First five polynomials u(n,x):
%e A210039 1
%e A210039 3
%e A210039 6 + x
%e A210039 10 + 5x
%e A210039 21 + 35x + 7x^2.
%t A210039 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210039 u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A210039 v[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
%t A210039 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210039 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210039 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210039 TableForm[cu]
%t A210039 Flatten[%]    (* A210039 *)
%t A210039 Table[Expand[v[n, x]], {n, 1, z}]
%t A210039 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210039 TableForm[cv]
%t A210039 Flatten[%]    (* A210040 *)
%t A210039 Table[u[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
%t A210039 Table[v[n, x] /. x -> 1, {n, 1, z}] (* A000225 *)
%Y A210039 Cf. A034839, A210040, A208510.
%K A210039 nonn,tabf
%O A210039 1,2
%A A210039 _Clark Kimberling_, Mar 17 2012