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A208514 Triangle of coefficients of polynomials u(n,x) jointly generated with A208515; see the Formula section. 4

%I

%S 1,1,1,1,2,2,1,3,4,3,1,4,6,7,5,1,5,8,12,13,8,1,6,10,18,24,23,13,1,7,

%T 12,25,38,46,41,21,1,8,14,33,55,78,88,72,34,1,9,16,42,75,120,158,165,

%U 126,55,1,10,18,52,98,173,255,313,307,219,89,1,11,20,63,124,238

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A208515; see the Formula section.

%C u(n,n) = Fibonacci(n), A000054

%C u(n+1,n) = A208354(n)

%C col 1: A000012

%C col 2: A000027

%C col 3: A005843

%C col 4: A055998

%C col 5: A140090

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

%F v(n,x)=x*u(n-1,x)+x*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...2...2

%e 1...3...4...3

%e 1...4...6...7...5

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

%e 1

%e 1 + x

%e 1 + 2x + 2x^2

%e 1 + 3x + 4x^2 + 3x^3

%e 1 + 4x + 6x^2 + 7x^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*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[%] (* A208514 *)

%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[%] (* A208515 *)

%Y Cf. A208515.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Feb 28 2012

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Last modified April 12 01:36 EDT 2021. Contains 342912 sequences. (Running on oeis4.)