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Triangle of coefficients of polynomials u(n,x) jointly generated with A105070; see Formula section.
3

%I #16 Jan 22 2020 17:37:24

%S 1,1,2,1,6,1,12,4,1,20,20,1,30,60,8,1,42,140,56,1,56,280,224,16,1,72,

%T 504,672,144,1,90,840,1680,720,32,1,110,1320,3696,2640,352,1,132,1980,

%U 7392,7920,2112,64,1,156,2860,13728,20592,9152,832,1,182,4004

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A105070; see Formula section.

%C Subtriangle of the triangle given by (1, 0, 1, 0, 0, 0, 0, 0, 0, ...) DELTA (0, 2, -2, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938. - _Philippe Deléham_, Apr 08 2012

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

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

%F From _Philippe Deléham_, Apr 08 2012: (Start)

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

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

%F T(n,k) = 2*T(n-1,k) - T(n-2,k) + 2*T(n-2,k-1), 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.

%F T(n,k) = A034839(n,k)*2^k = binomial(n,2*k)*2^k . (End)

%e First seven rows:

%e 1;

%e 1, 2;

%e 1, 6,

%e 1, 12, 4;

%e 1, 20, 20,

%e 1, 30, 60, 8;

%e 1, 42, 140, 56;

%e From _Philippe Deléham_, Apr 08 2012: (Start)

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

%e 1;

%e 1, 0;

%e 1, 2, 0;

%e 1, 6, 0, 0;

%e 1, 12, 4, 0, 0;

%e 1, 20, 20, 0, 0, 0;

%e 1, 30, 60, 8, 0, 0, 0;

%e 1, 42, 140, 56, 0, 0, 0, 0; (End)

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

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

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

%t Table[Factor[u[n, x]], {n, 1, z}]

%t Table[Factor[v[n, x]], {n, 1, z}]

%t cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];

%t TableForm[cu]

%t Flatten[%] (* A207536 *)

%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[%] (* A105070 *)

%Y Cf. A105070.

%K nonn,tabf

%O 1,3

%A _Clark Kimberling_, Feb 18 2012