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A208337 Triangle of coefficients of polynomials v(n,x) jointly generated with A208836; see the Formula section.

%I A208337 #23 Jan 05 2025 19:51:39
%S A208337 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 A208337 8,33,85,145,162,109,34,1,9,42,125,255,361,344,201,55,1,10,52,175,413,
%U A208337 701,850,707,365,89,1,11,63,236,630,1239,1806,1918,1416,655
%N A208337 Triangle of coefficients of polynomials v(n,x) jointly generated with A208836; see the Formula section.
%C A208337 coef. of x(n-1) in u(n,x): A000045(n), Fibonacci numbers
%C A208337 coef. of x(n-1) in v(n,x): A000045(n+1)
%C A208337 row sums, u(n,1):  A000129
%C A208337 row sums, v(n,1):  A001333
%C A208337 alternating row sums, u(n,-1): 1,0,1,0,1,0,1,0,1,0,...
%C A208337 alternating row sums, v(n,-1): 1,-1,1,-1,1,-1,1,-1,...
%C A208337 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 A208337 C. Kimberling, <a href="https://web.archive.org/web/2024*/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 A208337 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208337 v(n,x)=(x+1)*u(n-1,x)+x*v(n-1,x),
%F A208337 where u(1,x)=1, v(1,x)=1.
%F A208337 From _Philippe Deléham_, Apr 09 2012: (Start)
%F A208337 As DELTA-triangle T(n,k) with 0<=k<=n :
%F A208337 G.f.: (1-y*x+y*x^2-y^2*x^2)/(1-x-y*x-y^2*x^2).
%F A208337 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 A208337 G.f.: -(1+x*y)*x*y/(-1+x*y+x^2*y^2+x). - _R. J. Mathar_, Aug 11 2015
%e A208337 First five rows:
%e A208337   1
%e A208337   1...2
%e A208337   1...3...3
%e A208337   1...4...7....5
%e A208337   1...5...12...15...8
%e A208337 First five polynomials v(n,x):
%e A208337   1
%e A208337   1 + 2x
%e A208337   1 + 3x + 3x^2
%e A208337   1 + 4x + 7x^2 + 5x^3
%e A208337   1 + 5x + 12x^2 + 15x^3 + 8x^4
%e A208337 (1, 0, -1/2, 1/2, 0, 0, 0, ...) DELTA (0, 2, -1/2, -1/2, 0, 0, 0, ...) begins :
%e A208337   1
%e A208337   1, 0
%e A208337   1, 2, 0
%e A208337   1, 3, 3, 0
%e A208337   1, 4, 7, 5, 0
%e A208337   1, 5, 12, 15, 8, 0
%e A208337   1, 6, 18, 31, 30, 13, 0
%e A208337   1, 7, 25, 54, 73, 58, 21, 0 . _Philippe Deléham_, Apr 09 2012
%t A208337 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208337 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208337 v[n_, x_] := (x + 1)*u[n - 1, x] + x*v[n - 1, x];
%t A208337 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208337 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208337 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208337 TableForm[cu]
%t A208337 Flatten[%]    (* A208336 *)
%t A208337 Table[Expand[v[n, x]], {n, 1, z}]
%t A208337 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208337 TableForm[cv]
%t A208337 Flatten[%]    (* A208337 *)
%t A208337 Table[u[n, x] /. x -> 1, {n, 1, z}] (*u row sums*)
%t A208337 Table[v[n, x] /. x -> 1, {n, 1, z}] (*v row sums*)
%t A208337 Table[u[n, x] /. x -> -1, {n, 1, z}](*u alt. row sums*)
%t A208337 Table[v[n, x] /. x -> -1, {n, 1, z}](*v alt. row sums*)
%Y A208337 Cf. A208336.
%K A208337 nonn,tabl
%O A208337 1,3
%A A208337 _Clark Kimberling_, Feb 26 2012