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%I #12 Aug 12 2015 04:02:32
%S 1,2,2,3,7,4,4,17,21,8,5,34,68,55,16,6,60,174,225,137,32,7,97,384,705,
%T 674,327,64,8,147,763,1863,2489,1883,761,128,9,212,1400,4362,7640,
%U 8012,5016,1735,256,10,294,2412,9318,20542,27996,24144,12885,3897
%N Triangle of coefficients of polynomials v(n,x) jointly generated with A208761; see the Formula section.
%C Alternating row sums: 1,0,0,0,0,0,0,0,0,... For a discussion and guide to related arrays, see A208510.
%C As triangle T(n,k) with 0<=k<=n, it is (2, -1/2, 1/2, 0 0 0 0 0 0 0 ...) DELTA (2, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 04 2012
%C Row sums are in A055099. - _Philippe Deléham_, Mar 04 2012
%F u(n,x)=u(n-1,x)+2x*v(n-1,x),
%F v(n,x)=(x+1)*u(n-1,x)+(x+1)*v(n-1,x),
%F where u(1,x)=1, v(1,x)=1.
%F As triangle T(n,k) with 0<=k<=n : T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k) + T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = T(1,1) = 2 and T(n,k) = 0 if k>n or if k<0. - _Philippe Deléham_, Mar 04 2012
%F G.f.: (-1-x*y)*x*y/(-1+x*y+x^2*y+2*x^2*y^2+2*x-x^2). - _R. J. Mathar_, Aug 12 2015
%e First five rows:
%e 1
%e 2...2
%e 3...7....4
%e 4...17...21...8
%e 5...34...68...55...16
%e First five polynomials v(n,x):
%e 1
%e 2 + 2x
%e 3 + 7x + 4x^2
%e 4 + 17x + 22x^2 + 8x^3
%e 5 + 34x + 68x^2 + 55x^3 + 16x^4
%t u[1, x_] := 1; v[1, x_] := 1; z = 16;
%t u[n_, x_] := u[n - 1, x] + 2 x*v[n - 1, x];
%t v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1) 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[%] (* A208761 *)
%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[%] (* A208762 *)
%Y Cf. A208761, A208510.
%K nonn,tabl
%O 1,2
%A _Clark Kimberling_, Mar 03 2012
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