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A208331 Triangle of coefficients of polynomials v(n,x) jointly generated with A208330; see the Formula section.

%I A208331 #13 Sep 08 2013 19:59:30
%S A208331 1,1,3,1,6,5,1,9,15,11,1,12,30,44,21,1,15,50,110,105,43,1,18,75,220,
%T A208331 315,258,85,1,21,105,385,735,903,595,171,1,24,140,616,1470,2408,2380,
%U A208331 1368,341,1,27,180,924,2646,5418,7140,6156,3069,683,1,30,225
%N A208331 Triangle of coefficients of polynomials v(n,x) jointly generated with A208330; see the Formula section.
%C A208331 Subtriangle of the triangle given by (1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 18 2012
%F A208331 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208331 v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x),
%F A208331 where u(1,x)=1, v(1,x)=1.
%F A208331 T(n,k) = A001045(n+2)*binomial(n-1,k). - _Philippe Deléham_, Mar 18 2012
%F A208331 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(1,0) = T(2,0) = 1, T(2,1) = 3 and T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 18 2012
%e A208331 First five rows:
%e A208331 1
%e A208331 1...3
%e A208331 1...6...5
%e A208331 1...9...15...11
%e A208331 1...12...30...44...21
%e A208331 First five polynomials u(n,x):
%e A208331 1, 1 + 3x, 1 + 6x + 5x^2, 1 + 9x + 15x^2 + 11x^3, 1+12x + 30x^2 + 44x^3 + 21x^4.
%e A208331 (1, 0, 0, 1, 0, 0, ...) DELTA (0, 3, -4/3, -2/3, 0, 0, ...) begins :
%e A208331 1
%e A208331 1, 0
%e A208331 1, 3, 0
%e A208331 1, 6, 5, 0
%e A208331 1, 9, 15, 11, 0
%e A208331 1, 12, 30, 44, 21, 0. - _Philippe Deléham_, Mar 18 2012
%t A208331 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208331 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208331 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A208331 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208331 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208331 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208331 TableForm[cu]
%t A208331 Flatten[%]  (* A208330 *)
%t A208331 Table[Expand[v[n, x]], {n, 1, z}]
%t A208331 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208331 TableForm[cv]
%t A208331 Flatten[%]  (* A208331 *)
%Y A208331 Cf. A208330.
%K A208331 nonn,tabl
%O A208331 1,3
%A A208331 _Clark Kimberling_, Feb 26 2012