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A210204 Triangle of coefficients of polynomials v(n,x) jointly generated with A210203; see the Formula section.

%I A210204 #6 Jan 14 2022 21:25:09
%S A210204 1,3,2,7,8,2,15,24,12,2,31,64,48,16,2,63,160,160,80,20,2,127,384,480,
%T A210204 320,120,24,2,255,896,1344,1120,560,168,28,2,511,2048,3584,3584,2240,
%U A210204 896,224,32,2,1023,4608,9216,10752,8064,4032,1344,288,36,2,2047
%N A210204 Triangle of coefficients of polynomials v(n,x) jointly generated with A210203; see the Formula section.
%C A210204 Column 1:  -1+2^n.
%C A210204 Row sums: A048473.
%C A210204 Alternating row sums: 1,1,1,1,1,1,1,1,1,...
%C A210204 For a discussion and guide to related arrays, see A208510.
%C A210204 Row sums without first column give A056182. - _Alois P. Heinz_, Jan 14 2022
%F A210204 u(n,x)=u(n-1,x)+v(n-1,x)+1,
%F A210204 v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A210204 where u(1,x)=1, v(1,x)=1.
%e A210204 First five rows:
%e A210204 1
%e A210204 3....2
%e A210204 7....8....2
%e A210204 15...24...12...2
%e A210204 31...64...48...16...2
%e A210204 First three polynomials v(n,x): 1, 3 + 2x , 7 + 8x + 2x^2.
%t A210204 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A210204 u[n_, x_] := u[n - 1, x] + v[n - 1, x] + 1;
%t A210204 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A210204 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A210204 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A210204 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A210204 TableForm[cu]
%t A210204 Flatten[%]   (* A210203 *)
%t A210204 Table[Expand[v[n, x]], {n, 1, z}]
%t A210204 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A210204 TableForm[cv]
%t A210204 Flatten[%]   (* A210204 *)
%Y A210204 Cf. A210203, A208510.
%Y A210204 Cf. A056182.
%K A210204 nonn,tabl
%O A210204 1,2
%A A210204 _Clark Kimberling_, Mar 18 2012