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A208339 Triangle of coefficients of polynomials v(n,x) jointly generated with A208838; see the Formula section.

%I A208339 #16 Aug 11 2015 13:44:27
%S A208339 1,1,3,1,4,7,1,5,13,17,1,6,20,40,41,1,7,28,72,117,99,1,8,37,114,241,
%T A208339 332,239,1,9,47,167,425,769,921,577,1,10,58,232,682,1492,2368,2512,
%U A208339 1393,1,11,70,310,1026,2598,5008,7096,6761,3363,1,12,83,402,1472
%N A208339 Triangle of coefficients of polynomials v(n,x) jointly generated with A208838; see the Formula section.
%C A208339 Subtriangle of the triangle given by (1, 0, -2/3, 2/3, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -2/3, -1/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 27 2012
%F A208339 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208339 v(n,x)=(x+1)*u(n-1,x)+2x*v(n-1,x),
%F A208339 where u(1,x)=1, v(1,x)=1.
%F A208339 Contribution from _Philippe Deléham_, Mar 27 2012: (Start)
%F A208339 As DELTA-triangle T(n,k) with 0<=k<=n:
%F A208339 G.f.: (1-2*y*x+2*y*x^2-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).
%F A208339 T(n,k) = T(n-1,k) + 2*T(n-1,k-1) - T(n-2,k-1) + T(n-2,k-2), T(0,0) = T(1,0) = T(2,0) = 1, T(2,1) = 3, T(1,1) = T(2,2) = 0 nd T(n,k) = 0 if k<0 or if k>n. (End)
%F A208339 G.f.: -(1+x*y)*x*y/(-1+2*x*y-x^2*y+x^2*y^2+x). - _R. J. Mathar_, Aug 11 2015
%e A208339 First five rows:
%e A208339 1
%e A208339 1...3
%e A208339 1...4...7
%e A208339 1...5...13...17
%e A208339 1...6...20...40...41
%e A208339 First five polynomials v(n,x):
%e A208339 1
%e A208339 1 + 3x
%e A208339 1 + 4x + 7x^2
%e A208339 1 + 5x + 13x^2 + 17x^3
%e A208339 1 + 6x + 20x^2 + 40x^3 + 41x^4
%e A208339 Contribution from _Philippe Deléham_, Mar 27 2012: (Start)
%e A208339 (1, 0, -2/3, 2/3, 0, 0,...) DELTA (0, 3, -2/3, -1/3, 0, 0,...) begins :
%e A208339 1
%e A208339 1, 0
%e A208339 1, 3, 0
%e A208339 1, 4, 7, 0
%e A208339 1, 5, 13, 17, 0
%e A208339 1, 6, 20, 40, 41, 0. (End)
%t A208339 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208339 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208339 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A208339 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208339 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208339 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208339 TableForm[cu]
%t A208339 Flatten[%]   (* A208338 *)
%t A208339 Table[Expand[v[n, x]], {n, 1, z}]
%t A208339 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208339 TableForm[cv]
%t A208339 Flatten[%]   (* A208339 *)
%Y A208339 Cf. A208338.
%K A208339 nonn,tabl
%O A208339 1,3
%A A208339 _Clark Kimberling_, Feb 27 2012