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A210557 Triangle of coefficients of polynomials u(n,x) jointly generated with A210558; see the Formula section. 4

%I

%S 1,1,2,1,3,5,1,4,10,12,1,5,16,30,29,1,6,23,56,87,70,1,7,31,91,185,245,

%T 169,1,8,40,136,334,584,676,408,1,9,50,192,546,1158,1784,1836,985,1,

%U 10,61,260,834,2052,3850,5312,4925,2378,1,11,73,341,1212,3366

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210558; see the Formula section.

%C Row sums: powers of 3 (see A000244).

%C For a discussion and guide to related arrays, see A208510.

%C Subtriangle of (1, 0, -1/2, 1/2, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, 1/2, -1/2, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Mar 23 2012

%C Up to reflection at the vertical axis, this triangle coincides with the triangle given in A164981, i.e. the numbers are the same just read row-wise in the opposite direction. [_Christine Bessenrodt_, Jul 20 2012]

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

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

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

%F Contribution from _Philippe Deléham_, Mar 23 2012. (Start)

%F As DELTA-triangle T(n,k) with 0<=k<=n :

%F G.f.: (1-2*y*x+y*x^2-y^2*x^2)/(1-x-2*y*x+y*x^2-y^2*x^2).

%F 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) = 1, T(1,1) = T(2,2) = 0, T(2,1) = 2 and T(n,k) = 0 if k<0 or if k>n.

%e First five rows:

%e 1

%e 1...2

%e 1...3...5

%e 1...4...10...12

%e 1...5...16...30...29

%e First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 5x^2.

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

%e 1

%e 1, 0

%e 1, 2, 0

%e 1, 3, 5, 0

%e 1, 4, 10, 12, 0

%e 1, 5, 16, 30, 29, 0 . _Philippe Deléham_, Mar 23 2012

%t u[1, x_] := 1; v[1, x_] := 1; z = 16;

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

%t v[n_, x_] := 2 x*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;

%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[%] (* A210557 *)

%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[%] (* A210558 *)

%Y Cf. A210558, A208510, A164981.

%K nonn,tabl

%O 1,3

%A _Clark Kimberling_, Mar 22 2012

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Last modified November 11 16:07 EST 2019. Contains 329019 sequences. (Running on oeis4.)