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A209169 Triangle of coefficients of polynomials v(n,x) jointly generated with A209168; see the Formula section.

%I A209169 #18 Aug 07 2025 04:35:07
%S A209169 1,2,3,3,7,7,5,16,23,17,8,33,65,70,41,13,65,159,233,204,99,21,124,362,
%T A209169 654,776,577,239,34,231,782,1676,2447,2461,1597,577,55,423,1627,4018,
%U A209169 6937,8586,7534,4348,1393,89,764,3289,9179,18202,26597,28750
%N A209169 Triangle of coefficients of polynomials v(n,x) jointly generated with A209168; see the Formula section.
%C A209169 Column 1: Fibonacci numbers (A000045).
%C A209169 For a discussion and guide to related arrays, see A208510.
%F A209169 u(n,x)=u(n-1,x)+(x+1)*v(n-1,x),
%F A209169 v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x)+1,
%F A209169 where u(1,x)=1, v(1,x)=1.
%F A209169 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-2), T(1,0) = 1, T(2,0) = 2, T(2,1) = 3, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 11 2012
%e A209169 First five rows:
%e A209169   1
%e A209169   2   3
%e A209169   3   7    7
%e A209169   5   16   23   17
%e A209169   8   33   65   70   41
%e A209169 First three polynomials v(n,x): 1, 2 + 3*x, 3 + 7*x + 7*x^2.
%t A209169 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209169 u[n_, x_] := u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209169 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
%t A209169 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209169 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209169 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209169 TableForm[cu]
%t A209169 Flatten[%]    (* A209168 *)
%t A209169 Table[Expand[v[n, x]], {n, 1, z}]
%t A209169 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209169 TableForm[cv]
%t A209169 Flatten[%]    (* A209169 *)
%Y A209169 Cf. A209168, A208510.
%K A209169 nonn,tabl
%O A209169 1,2
%A A209169 _Clark Kimberling_, Mar 08 2012