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A208328 Triangle of coefficients of polynomials u(n,x) jointly generated with A208329; see the Formula section.

%I A208328 #17 Jan 22 2020 20:12:51
%S A208328 1,1,1,1,1,3,1,1,5,5,1,1,7,9,11,1,1,9,13,25,21,1,1,11,17,43,53,43,1,1,
%T A208328 13,21,65,97,125,85,1,1,15,25,91,153,255,273,171,1,1,17,29,121,221,
%U A208328 441,597,609,341,1,1,19,33,155,301,691,1089,1443,1325,683,1,1,21
%N A208328 Triangle of coefficients of polynomials u(n,x) jointly generated with A208329; see the Formula section.
%C A208328 Row sums, u(n,1):  A000129
%C A208328 Row sums, v(n,1):  A001333
%C A208328 Subtriangle of the triangle T(n,k) given by (1, 0, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 07 2012
%F A208328 u(n,x) = u(n-1,x) + x*v(n-1,x),
%F A208328 v(n,x) = 2x*u(n-1,x) + x*v(n-1,x),
%F A208328 where u(1,x)=1, v(1,x)=1.
%F A208328 From _Philippe Deléham_, Mar 07 2012: (Start)
%F A208328 As DELTA-triangle T(n,k), 0 <= k <= n:
%F A208328 G.f.: (1-y*x - y*(2*y-1)*x^2)/(1-(1+y)*x-y(2*y-1)*x^2).
%F A208328 T(n,k) = T(n-1,k) + T(n-1,k-1) - T(n-2,k-1) + 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 1, T(1,1) = 0, T(n,k) = 0 if k < 0 or if k > n.
%F A208328 Sum_{k=0..n, n>0} T(n,k)*x^k = A000012(n), A000129(n), A083858(n) for x = 0, 1, 2 respectively. (End)
%e A208328 First five rows:
%e A208328   1;
%e A208328   1, 1;
%e A208328   1, 1, 3;
%e A208328   1, 1, 5, 5;
%e A208328   1, 1, 7, 9, 11;
%e A208328 First five polynomials u(n,x):
%e A208328   1
%e A208328   1 + x
%e A208328   1 + x + 3x^2
%e A208328   1 + x + 5x^2 + 5x^3
%e A208328   1 + x + 7x^2 + 9x^3 + 11x^4.
%e A208328 From _Philippe Deléham_, Mar 07 2012: (Start)
%e A208328 (1, 0, -1, 1, 0, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, 0, ...) begins:
%e A208328   1;
%e A208328   1, 0;
%e A208328   1, 1,  0;
%e A208328   1, 1,  3,  0;
%e A208328   1, 1,  5,  5,  0;
%e A208328   1, 1,  7,  9, 11,  0;
%e A208328   1, 1,  9, 13, 25, 21,  0;
%e A208328   1, 1, 11, 17, 43, 53, 43, 0; (End)
%t A208328 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208328 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208328 v[n_, x_] := 2 x*u[n - 1, x] + x*v[n - 1, x];
%t A208328 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208328 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208328 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208328 TableForm[cu]
%t A208328 Flatten[%]  (* A208328 *)
%t A208328 Table[Expand[v[n, x]], {n, 1, z}]
%t A208328 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208328 TableForm[cv]
%t A208328 Flatten[%]  (* A208329 *)
%Y A208328 Cf. A208329.
%K A208328 nonn,tabl
%O A208328 1,6
%A A208328 _Clark Kimberling_, Feb 26 2012