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Triangle of coefficients of polynomials v(n,x) jointly generated with A208836; see the Formula section.
5

%I #22 Jan 21 2019 06:10:34

%S 1,1,2,1,3,3,1,4,7,5,1,5,12,15,8,1,6,18,31,30,13,1,7,25,54,73,58,21,1,

%T 8,33,85,145,162,109,34,1,9,42,125,255,361,344,201,55,1,10,52,175,413,

%U 701,850,707,365,89,1,11,63,236,630,1239,1806,1918,1416,655

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208836; see the Formula section.

%C coef. of x(n-1) in u(n,x): A000045(n), Fibonacci numbers

%C coef. of x(n-1) in v(n,x): A000045(n+1)

%C row sums, u(n,1): A000129

%C row sums, v(n,1): A001333

%C alternating row sums, u(n,-1): 1,0,1,0,1,0,1,0,1,0,...

%C alternating row sums, v(n,-1): 1,-1,1,-1,1,-1,1,-1,...

%C Subtriangle of the triangle given by (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 09 2012

%H C. Kimberling, <a href="https://www.fq.math.ca/Scanned/39-5/kimberling.pdf">Enumeration of paths, compositions of integers and Fibonacci numbers</a>, Fib. Quarterly 39 (5) (2001) 430-435 Figure 2.

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

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

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

%F From _Philippe Deléham_, Apr 09 2012: (Start)

%F As DELTA-triangle T(n,k) with 0<=k<=n :

%F G.f.: (1-y*x+y*x^2-y^2*x^2)/(1-x-y*x-y^2*x^2).

%F T(n,k) = T(n-1,k) + T(n-1,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 2, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k<0 or if k>n. (End)

%F G.f.: -(1+x*y)*x*y/(-1+x*y+x^2*y^2+x). - _R. J. Mathar_, Aug 11 2015

%e First five rows:

%e 1

%e 1...2

%e 1...3...3

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

%e 1...5...12...15...8

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

%e 1

%e 1 + 2x

%e 1 + 3x + 3x^2

%e 1 + 4x + 7x^2 + 5x^3

%e 1 + 5x + 12x^2 + 15x^3 + 8x^4

%e (1, 0, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, ...) begins :

%e 1

%e 1, 0

%e 1, 2, 0

%e 1, 3, 3, 0

%e 1, 4, 7, 5, 0

%e 1, 5, 12, 15, 8, 0

%e 1, 6, 18, 31, 30, 13, 0

%e 1, 7, 25, 54, 73, 58, 21, 0 . _Philippe Deléham_, Apr 09 2012

%t u[1, x_] := 1; v[1, x_] := 1; z = 13;

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

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

%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[%] (* A208336 *)

%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[%] (* A208337 *)

%t Table[u[n, x] /. x -> 1, {n, 1, z}] (*u row sums*)

%t Table[v[n, x] /. x -> 1, {n, 1, z}] (*v row sums*)

%t Table[u[n, x] /. x -> -1, {n, 1, z}](*u alt. row sums*)

%t Table[v[n, x] /. x -> -1, {n, 1, z}](*v alt. row sums*)

%Y Cf. A208336.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Feb 26 2012