A210756 Triangle of coefficients of polynomials v(n,x) jointly generated with A210755; see the Formula section.

1, 3, 2, 6, 10, 5, 10, 30, 33, 12, 15, 70, 127, 100, 29, 21, 140, 371, 472, 291, 70, 28, 252, 910, 1656, 1624, 822, 169, 36, 420, 1974, 4800, 6640, 5294, 2273, 408, 45, 660, 3906, 12144, 22166, 24702, 16589, 6184, 985, 55, 990, 7194, 27720, 63954

Offset: 1

Author

Clark Kimberling, Mar 25 2012

Keywords

• nonn
• tabl

Comments

Column 1: triangular numbers, A000217

Coefficient of v(n,x): A000129(n)

Row sums: A002450

Alternating row sums: 1,1,1,1,1,1,1,1,1,...

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

Examples

			First five rows:
1
3....2
6....10...5
10...30...33....12
15...70...127...100...29
First three polynomials v(n,x): 1, 3 + 2x, 6 + 10x + 5x^2
		

Crossrefs

Cf. A210755, A208510.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := (x + 1)*u[n - 1, x] + 2 x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + (x + 1)*v[n - 1, x] + 1;
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[%]    (* A210755 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210756 *)
Table[u[n, x] /. x -> 1, {n, 1, z}] (* A002450 *)
Table[v[n, x] /. x -> 1, {n, 1, z}] (* A002450 *)

Formula

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

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

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