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A208338 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section.

%I A208338 #14 Jan 24 2020 03:27:17
%S A208338 1,1,1,1,2,3,1,3,7,7,1,4,12,20,17,1,5,18,40,57,41,1,6,25,68,129,158,
%T A208338 99,1,7,33,105,243,399,431,239,1,8,42,152,410,824,1200,1160,577,1,9,
%U A208338 52,210,642,1506,2692,3528,3089,1393,1,10,63,280,952,2532,5290
%N A208338 Triangle of coefficients of polynomials u(n,x) jointly generated with A208339; see the Formula section.
%C A208338 Subtriangle of the triangle given by (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 09 2012
%F A208338 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A208338 v(n,x) = (x+1)*u(n-1,x) + 2x*v(n-1,x),
%F A208338 where u(1,x)=1, v(1,x)=1.
%F A208338 From _Philippe Deléham_, Apr 09 2012: (Start)
%F A208338 As DELTA-triangle T(n,k) with 0 <= k <= n:
%F A208338 G.f.: (1-2*y*x-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).
%F A208338 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) = T(2,1) = 1, T(1,1) = T(2,2) = 0 and T(n,k) = 0 if k < 0 or if k > n. (End)
%e A208338 First five rows:
%e A208338   1;
%e A208338   1,  1;
%e A208338   1,  2,  3;
%e A208338   1,  3,  7,  7;
%e A208338   1,  4, 12, 20, 17;
%e A208338 First five polynomials u(n,x):
%e A208338   1
%e A208338   1 +  x
%e A208338   1 + 2x +  3x^2
%e A208338   1 + 3x +  7x^2 +  7x^3
%e A208338   1 + 4x + 12x^2 + 20x^3 + 17x^4
%e A208338 From _Philippe Deléham_, Apr 09 2012: (Start)
%e A208338 (1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -1, 0, 0, 0, ...) begins:
%e A208338   1;
%e A208338   1,  0;
%e A208338   1,  1,  0;
%e A208338   1,  2,  3,  0;
%e A208338   1,  3,  7,  7,  0;
%e A208338   1,  4, 12, 20, 17,  0;
%e A208338   1,  5, 18, 40, 57, 41,  0; (End)
%t A208338 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208338 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208338 v[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x];
%t A208338 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208338 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208338 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208338 TableForm[cu]
%t A208338 Flatten[%]   (* A208338 *)
%t A208338 Table[Expand[v[n, x]], {n, 1, z}]
%t A208338 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208338 TableForm[cv]
%t A208338 Flatten[%]   (* A208339 *)
%Y A208338 Cf. A208339.
%K A208338 nonn,tabl
%O A208338 1,5
%A A208338 _Clark Kimberling_, Feb 27 2012