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%I #6 Oct 17 2012 10:05:03
%S 1,3,1,7,4,15,12,1,31,32,6,63,80,24,1,127,192,80,8,255,448,240,40,1,
%T 511,1024,672,160,10,1023,2304,1792,560,60,1,2047,5120,4608,1792,280,
%U 12,4095,11264,11520,5376,1120,84,1,8191,24576,28160,15360,4032
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210197; see the Formula section.
%C Row sums: A005409
%C Column 1: -1+2^n
%C Alternating row sums: 1, 2,3,4,5,6,..., A000027
%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+1)*u(n-1,x)+v(n-1,x)+1,
%F where u(1,x)=1, v(1,x)=1.
%e First five rows:
%e 1
%e 3....1
%e 15...12...1
%e 31...32...6
%e 63...80...24...1
%e First three polynomials v(n,x): 1, 3 + x , 15 + 12x + x^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 + 1)*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[%] (* A210197 *)
%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[%] (* A210198 *)
%t Table[u[n, x] /. x -> 1, {n, 1, z}] (* A048739 *)
%t Table[v[n, x] /. x -> 1, {n, 1, z}] (* A005409 *)
%t Table[u[n, x] /. x -> -1, {n, 1, z}] (* A000217 *)
%t Table[v[n, x] /. x -> -1, {n, 1, z}] (* A000027 *)
%Y Cf. A210197, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 18 2012
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