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Triangle of coefficients of polynomials v(n,x) jointly generated with A210860; see the Formula section.
3

%I #9 Dec 17 2012 15:43:03

%S 1,2,2,6,7,3,16,30,20,5,50,116,108,47,8,156,460,552,338,105,13,532,

%T 1842,2692,2119,941,221,21,1856,7532,13072,12574,7216,2452,451,34,

%U 6876,31600,63240,71860,50525,22371,6035,895,55,26200,135576,308568

%N Triangle of coefficients of polynomials v(n,x) jointly generated with A210860; see the Formula section.

%C Row n ends with F(n+1), where F=A000045 (Fibonacci numbers).

%C Column 1: A013989

%C Alternating row sums: 1,0,2,1,3,2,4,3,5,4,...

%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....2

%e 6....7.....3

%e 16...30....20....5

%e 50...116...108...47...8

%e First three polynomials v(n,x): 1, 2 + 2x, 6 + 7x + 3x^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. A210860, A208510.

%K nonn,tabl

%O 1,2

%A _Clark Kimberling_, Mar 28 2012