A210863 Triangle of coefficients of polynomials v(n,x) jointly generated with A210862; see the Formula section.

1, 1, 2, 4, 5, 3, 9, 13, 13, 5, 16, 37, 47, 32, 8, 25, 96, 152, 147, 73, 13, 36, 217, 469, 587, 432, 158, 21, 49, 435, 1344, 2127, 2090, 1183, 330, 34, 64, 793, 3487, 7186, 8979, 6965, 3064, 669, 55, 81, 1342, 8179, 22544, 35296, 35304, 21754, 7577

Offset: 1

Author

Clark Kimberling, Mar 28 2012

Keywords

• nonn
• tabl

Comments

For n>1, row n starts with (n-1)^2 and ends with F(n+1), where F=A000045 (Fibonacci numbers).

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

Examples

			First five rows:
1
1....2
4....5....3
9....13...13...5
16...37...47...32...8
First three polynomials v(n,x): 1, 1 + 2x, 4 + 5x + 3x^2
		

Crossrefs

Cf. A210862, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + n - 1)*u[n - 1, x] + x*v[n - 1, x];
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]   (* A210862 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]   (* A210863 *)

Formula

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

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

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