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 A208330 Triangle of coefficients of polynomials u(n,x) jointly generated with A208331; see the Formula section. 3

%I

%S 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 43,1,7,63,175,385,441,301,85,1,8,84,280,770,1176,1204,680,171,1,9,

%U 108,420,1386,2646,3612,3060,1539,341,1,10,135,600,2310,5292,9030

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

%C 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 u(n,x)=u(n-1,x)+x*v(n-1,x),

%F v(n,x)=2x*u(n-1,x)+(x+1)*v(n-1,x),

%F where u(1,x)=1, v(1,x)=1.

%F T(n,k) = A001045(k+1)*binomial(n-1,k). - _Philippe Deléham_, Mar 18 2012

%F 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 First five rows:

%e 1

%e 1...1

%e 1...2...3

%e 1...3...9....5

%e 1...4...18...20...11

%e First five polynomials u(n,x):

%e 1, 1 + x, 1 + 2x + 3x^2, 1 + 3x + 9x^2 + 5x^3, 1 + 4x + 18x^2 + 20x^3 + 11x^4.

%e (1, 0, 0, 1, 0, 0, ...) DELTA (0, 1, 2, -2, 0, 0, ...) begins :

%e 1

%e 1, 0

%e 1, 1, 0

%e 1, 2, 3, 0

%e 1, 3, 9, 5, 0

%e 1, 4, 18, 20, 11, 0

%e 1, 5, 30, 50, 55, 21, 0. - _Philippe Deléham_, Mar 18 2012

%t u[1, x_] := 1; v[1, x_] := 1; z = 13;

%t u[n_, x_] := u[n - 1, x] + x*v[n - 1, x];

%t v[n_, x_] := 2 x*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[%] (* A208330 *)

%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[%] (* A208331 *)

%Y Cf. A208331, A208510.

%K nonn,tabl

%O 1,5

%A _Clark Kimberling_, Feb 26 2012

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