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

%I A208330 #13 Sep 08 2013 19:59:30
%S A208330 1,1,1,1,2,3,1,3,9,5,1,4,18,20,11,1,5,30,50,55,21,1,6,45,100,165,126,
%T A208330 43,1,7,63,175,385,441,301,85,1,8,84,280,770,1176,1204,680,171,1,9,
%U A208330 108,420,1386,2646,3612,3060,1539,341,1,10,135,600,2310,5292,9030
%N A208330 Triangle of coefficients of polynomials u(n,x) jointly generated with A208331; see the Formula section.
%C A208330 Subtriangle of the triangle given by (1, 0, 0, 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 18 2012
%F A208330 u(n,x)=u(n-1,x)+x*v(n-1,x),
%F A208330 v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x),
%F A208330 where u(1,x)=1, v(1,x)=1.
%F A208330 T(n,k) = A001045(k+1)*binomial(n-1,k). - _Philippe Deléham_, Mar 18 2012
%F A208330 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) = T(2,1) = 1, T(n,k) = 0 if k<0 or if k>=n. - _Philippe Deléham_, Mar 18 2012
%e A208330 First five rows:
%e A208330 1
%e A208330 1...1
%e A208330 1...2...3
%e A208330 1...3...9....5
%e A208330 1...4...18...20...11
%e A208330 First five polynomials u(n,x):
%e A208330 1, 1 + x, 1 + 2x + 3x^2, 1 + 3x + 9x^2 + 5x^3, 1 + 4x + 18x^2 + 20x^3 + 11x^4.
%e A208330 (1, 0, 0, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins :
%e A208330 1
%e A208330 1, 0
%e A208330 1, 1, 0
%e A208330 1, 2, 3, 0
%e A208330 1, 3, 9, 5, 0
%e A208330 1, 4, 18, 20, 11, 0
%e A208330 1, 5, 30, 50, 55, 21, 0. - _Philippe Deléham_, Mar 18 2012
%t A208330 u[1, x_] := 1; v[1, x_] := 1; z = 13;
%t A208330 u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];
%t A208330 v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x];
%t A208330 Table[Expand[u[n, x]], {n, 1, z/2}]
%t A208330 Table[Expand[v[n, x]], {n, 1, z/2}]
%t A208330 cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
%t A208330 TableForm[cu]
%t A208330 Flatten[%]  (* A208330 *)
%t A208330 Table[Expand[v[n, x]], {n, 1, z}]
%t A208330 cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
%t A208330 TableForm[cv]
%t A208330 Flatten[%]  (* A208331 *)
%Y A208330 Cf. A208331, A208510.
%K A208330 nonn,tabl
%O A208330 1,5
%A A208330 _Clark Kimberling_, Feb 26 2012