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

1, 2, 3, 3, 11, 7, 4, 26, 38, 17, 5, 50, 124, 121, 41, 6, 85, 314, 499, 362, 99, 7, 133, 679, 1555, 1805, 1043, 239, 8, 196, 1316, 4054, 6672, 6096, 2926, 577, 9, 276, 2352, 9318, 20326, 26048, 19610, 8049, 1393, 10, 375, 3948, 19482, 53932, 90706

Offset: 1

Author

Clark Kimberling, Mar 25 2012

Keywords

• nonn
• tabl

Comments

Row n starts with n and ends with A001333(n).

Column 2: A051925

Row sums: A002450

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

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

Examples

			First five rows:
1
2...3
3...11...7
4...26...38....17
5...50...124...121...41
First three polynomials u(n,x): 1, 2 + 3x, 3 + 11x + 7x^2.
		

Crossrefs

Cf. A210754, 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.