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A209145 Triangle of coefficients of polynomials u(n,x) jointly generated with A122075; see the Formula section.

%I A209145 #8 Aug 06 2025 08:41:48
%S A209145 1,2,1,4,4,1,7,10,5,1,12,22,16,6,1,20,45,43,23,7,1,33,88,104,72,31,8,
%T A209145 1,54,167,235,199,110,40,9,1,88,310,506,506,340,158,50,10,1,143,566,
%U A209145 1051,1211,956,538,217,61,11,1,232,1020,2123,2768,2507,1652,805,288,73,12,1
%N A209145 Triangle of coefficients of polynomials u(n,x) jointly generated with A122075; see the Formula section.
%C A209145 For a discussion and guide to related arrays, see A208510.
%F A209145 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A209145 v(n,x)=u(n-1,x)+x*v(n-1,x)+1,
%F A209145 where u(1,x)=1, v(1,x)=1.
%e A209145 First five rows:
%e A209145   1
%e A209145   2    1
%e A209145   4    4    1
%e A209145   7    10   5    1
%e A209145   12   22   16   6   1
%e A209145 First three polynomials u(n,x): 1, 2 + x, 4 + 4*x + x^2.
%t A209145 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209145 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209145 v[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
%t A209145 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209145 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209145 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209145 TableForm[cu]
%t A209145 Flatten[%]    (* A209145 *)
%t A209145 Table[Expand[v[n, x]], {n, 1, z}]
%t A209145 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209145 TableForm[cv]
%t A209145 Flatten[%]    (* A122075 *)
%Y A209145 Cf. A209144, A208510.
%K A209145 nonn,tabl
%O A209145 1,2
%A A209145 _Clark Kimberling_, Mar 06 2012