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Triangle of coefficients of polynomials u(n,x) jointly generated with A210861; see the Formula section of A210861.
3

%I #18 Jan 12 2022 09:30:50

%S 1,2,1,4,5,2,10,18,12,3,26,64,62,28,5,76,230,286,183,60,8,232,846,

%T 1298,1073,503,126,13,764,3220,5832,5884,3563,1288,255,21,2620,12608,

%U 26436,31530,23353,10956,3158,506,34,9496,51084,121276,166630

%N Triangle of coefficients of polynomials u(n,x) jointly generated with A210861; see the Formula section of A210861.

%C Row n ends with F(n), where F=A000045 (Fibonacci numbers).

%C Column 1: A000085

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

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

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

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

%e First five rows:

%e 1

%e 2 1

%e 4 5 2

%e 10 18 12 3

%e 26 64 62 28 5

%e First three polynomials u(n,x): 1, 2 + x, 4 + 5x + 2x^2.

%t u[1, x_] := 1; v[1, x_] := 1; z = 14;

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

%t v[n_, x_] := (x + n)*u[n - 1, x] + x*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[%] (* A210860 *)

%t cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

%t TableForm[cv]

%t Flatten[%] (* A210861 *)

%Y Cf. A210861, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 28 2012

%E Definition clarified by _Alonso del Arte_ and _Harvey P. Dale_, Dec 17 2012