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A209746 Triangle of coefficients of polynomials v(n,x) jointly generated with A209745; see the Formula section.

%I A209746 #17 Jan 24 2020 03:28:28
%S A209746 1,2,2,3,7,4,5,17,20,8,8,37,65,52,16,13,75,176,210,128,32,21,146,428,
%T A209746 679,616,304,64,34,276,971,1921,2312,1696,704,128,55,511,2097,4970,
%U A209746 7449,7240,4464,1600,256,89,931,4366,12056,21622,26146,21344
%N A209746 Triangle of coefficients of polynomials v(n,x) jointly generated with A209745; see the Formula section.
%C A209746 Row n begins with F(n+1) and ends with 2^(n-1), where F=A000045 (Fibonacci numbers).
%C A209746 Alternating row sums: 1,0,0,0,0,0,0,0,0,0,...
%C A209746 For a discussion and guide to related arrays, see A208510.
%C A209746 Riordan array ((1+x)/(1-x-x^2), (2x+x^2)/(1-x-x^2)). - _Philippe Deléham_, Mar 24 2012
%C A209746 Triangle given by (2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 26 2012
%F A209746 u(n,x) = x*u(n-1,x) + (x+1)*v(n-1,x),
%F A209746 v(n,x) = (x+1)*u(n-1,x) + (x+1)*v(n-1,x),
%F A209746 where u(1,x)=1, v(1,x)=1.
%F A209746 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k) + T(n-2,k-1), T(1,0) = 1, T(2,0) = T(2,1) = 2, T(n,k) = 0 if k < 0 or if k >= 0. - _Philippe Deléham_, Mar 24 2012
%e A209746 First five rows:
%e A209746   1;
%e A209746   2,  2;
%e A209746   3,  7,  4;
%e A209746   5, 17, 20,  8;
%e A209746   8, 37, 65, 52, 16;
%e A209746 First three polynomials v(n,x):
%e A209746   1
%e A209746   2 + 2x
%e A209746   3 + 7x + 4x^2.
%t A209746 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A209746 u[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209746 v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A209746 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A209746 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A209746 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A209746 TableForm[cu]
%t A209746 Flatten[%]    (* A209745 *)
%t A209746 Table[Expand[v[n, x]], {n, 1, z}]
%t A209746 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A209746 TableForm[cv]
%t A209746 Flatten[%]    (* A209746 *)
%Y A209746 Cf. A000045, A000079, A209745, A208510.
%K A209746 nonn,tabl
%O A209746 1,2
%A A209746 _Clark Kimberling_, Mar 13 2012