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

%I A208518 #12 Mar 30 2012 18:58:13
%S A208518 1,1,1,1,3,2,1,6,7,3,1,10,16,14,5,1,15,30,40,28,8,1,21,50,90,93,53,13,
%T A208518 1,28,77,175,238,203,99,21,1,36,112,308,518,588,428,181,34,1,45,156,
%U A208518 504,1008,1428,1380,873,327,55,1,55,210,780,1806,3066,3690,3105
%N A208518 Triangle of coefficients of polynomials u(n,x) jointly generated with A208519; see the Formula section.
%C A208518 coefficient of x^(n-1): = Fibonacci(n) = A000045(n)
%C A208518 col 1:  A000012
%C A208518 col 2:  A000217 (triangular numbers)
%C A208518 col 3:  A005581
%C A208518 col 4:  A117662
%C A208518 alternating row sums: signed version of (-1+Fibonacci(n))
%F A208518 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208518 v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,
%F A208518 where u(1,x)=1, v(1,x)=1.
%e A208518 First five rows:
%e A208518 1
%e A208518 1...1
%e A208518 1...3....2
%e A208518 1...6....7....3
%e A208518 1...10...16...14...5
%e A208518 First five polynomials u(n,x):
%e A208518 1
%e A208518 1 + x
%e A208518 1 + 3x + 2x^2
%e A208518 1 + 6x + 7x^2 + 3x^3
%e A208518 1 + 10x + 16x^2 + 14x^3 + 5x^4
%t A208518 u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t A208518 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208518 v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
%t A208518 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208518 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208518 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208518 TableForm[cu]
%t A208518 Flatten[%]   (* A208518 *)
%t A208518 Table[Expand[v[n, x]], {n, 1, z}]
%t A208518 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208518 TableForm[cv]
%t A208518 Flatten[%]   (* A208519 *)
%Y A208518 Cf. A208519.
%K A208518 nonn,tabl
%O A208518 1,5
%A A208518 _Clark Kimberling_, Feb 28 2012