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A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.

%I A210754 #13 May 13 2017 14:18:47
%S A210754 1,3,2,6,9,4,10,25,24,8,15,55,85,60,16,21,105,231,258,144,32,28,182,
%T A210754 532,833,728,336,64,36,294,1092,2241,2720,1952,768,128,45,450,2058,
%U A210754 5301,8361,8280,5040,1728,256,55,660,3630,11385,22363,28610,23920
%N A210754 Triangle of coefficients of polynomials v(n,x) jointly generated with A210753; see the Formula section.
%C A210754 Column 1: triangular numbers, A000217
%C A210754 Coefficient of v(n,x):  2^(n-1)
%C A210754 Row sums:  A035344
%C A210754 Alternating row sums: 1,1,1,1,1,1,1,1,1,...
%C A210754 For a discussion and guide to related arrays, see A208510.
%C A210754 Appears to be the reversed row polynomials of A165241 with the unit diagonal removed. If so, the o.g.f. is [1-(1+y)x]/[1-2(1+y)x+(1+y)x^2] - 1/(1-x) and the triangular matrix here may be formed by adding each column of the matrix of A056242, presented in the example section with the additional zeros, to its subsequent column with the first row ignored. - _Tom Copeland_, Jan 09 2017
%F A210754 u(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x)+1,
%F A210754 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A210754 where u(1,x)=1, v(1,x)=1.
%e A210754 First five rows:
%e A210754 1
%e A210754 3....2
%e A210754 6....9....4
%e A210754 10...25...24...8
%e A210754 15...55...85...60...16
%e A210754 First three polynomials v(n,x): 1, 3 + 2x, 6 + 9x +4x^2
%t A210754 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210754 u[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x] + 1;
%t A210754 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A210754 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210754 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210754 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210754 TableForm[cu]
%t A210754 Flatten[%]    (* A210753 *)
%t A210754 Table[Expand[v[n, x]], {n, 1, z}]
%t A210754 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210754 TableForm[cv]
%t A210754 Flatten[%]    (* A210754 *)
%t A210754 Table[u[n, x] /. x -> 1, {n, 1, z}] (* A007070 *)
%t A210754 Table[v[n, x] /. x -> 1, {n, 1, z}] (* A035344 *)
%Y A210754 Cf. A210753, A208510.
%Y A210754 Cf. A056242, A165241.
%K A210754 nonn,tabl
%O A210754 1,2
%A A210754 _Clark Kimberling_, Mar 25 2012