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A208519 Triangle of coefficients of polynomials v(n,x) jointly generated with A208518; see the Formula section.

%I A208519 #9 Mar 30 2012 18:58:13
%S A208519 1,2,2,3,5,3,4,9,11,5,5,14,26,23,8,6,20,50,65,45,13,7,27,85,145,150,
%T A208519 86,21,8,35,133,280,385,329,160,34,9,44,196,490,840,952,692,293,55,10,
%U A208519 54,276,798,1638,2310,2232,1413,529,89,11,65,375,1230,2940,4956
%N A208519 Triangle of coefficients of polynomials v(n,x) jointly generated with A208518; see the Formula section.
%C A208519 coefficient of x^(n-1): Fibonacci(n+1) = A000045(n+1)
%C A208519 col 1:  A000027
%C A208519 col 2:  A000096
%C A208519 col 3:  A051925
%C A208519 row sums:  A002878 (bisection of Lucas sequence)
%C A208519 alternating row sums:  A000045(n-2), Fibonacci numbers
%F A208519 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208519 v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A208519 where u(1,x)=1, v(1,x)=1.
%e A208519 First five rows:
%e A208519 1
%e A208519 2...2
%e A208519 3...5....3
%e A208519 4...9....11...5
%e A208519 5...14...26...23...8
%e A208519 First five polynomials v(n,x):
%e A208519 1
%e A208519 2 + 2x
%e A208519 3 + 5x + 3x^2
%e A208519 4 + 9x + 11x^2 + 5x^3
%e A208519 5 + 14x + 26x^2 + 23x^3 + 8x^4
%t A208519 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208519 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208519 v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A208519 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208519 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208519 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208519 TableForm[cu]
%t A208519 Flatten[%]   (* A208518 *)
%t A208519 Table[Expand[v[n, x]], {n, 1, z}]
%t A208519 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208519 TableForm[cv]
%t A208519 Flatten[%]   (* A208519 *)
%Y A208519 Cf. A208518.
%K A208519 nonn,tabl
%O A208519 1,2
%A A208519 _Clark Kimberling_, Feb 28 2012