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 A208518 Triangle of coefficients of polynomials u(n,x) jointly generated with A208519; see the Formula section. 3

%I

%S 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 1,28,77,175,238,203,99,21,1,36,112,308,518,588,428,181,34,1,45,156,

%U 504,1008,1428,1380,873,327,55,1,55,210,780,1806,3066,3690,3105

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

%C coefficient of x^(n-1): = Fibonacci(n) = A000045(n)

%C col 1: A000012

%C col 2: A000217 (triangular numbers)

%C col 3: A005581

%C col 4: A117662

%C alternating row sums: signed version of (-1+Fibonacci(n))

%F u(n,x)=u(n-1,x)+x*v(n-1,x),

%F v(n,x)=x*u(n-1,x)+(x+1)*v(n-1,x)+1,

%F where u(1,x)=1, v(1,x)=1.

%e First five rows:

%e 1

%e 1...1

%e 1...3....2

%e 1...6....7....3

%e 1...10...16...14...5

%e First five polynomials u(n,x):

%e 1

%e 1 + x

%e 1 + 3x + 2x^2

%e 1 + 6x + 7x^2 + 3x^3

%e 1 + 10x + 16x^2 + 14x^3 + 5x^4

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

%t v[n_, x_] := x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

%t Table[Expand[u[n, x]], {n, 1, z/2}]

%t Table[Expand[v[n, x]], {n, 1, z/2}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A208518 *)

%t Table[Expand[v[n, x]], {n, 1, z}]

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A208519 *)

%Y Cf. A208519.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Feb 28 2012

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Last modified April 23 07:37 EDT 2021. Contains 343201 sequences. (Running on oeis4.)