A210562 Triangle of coefficients of polynomials v(n,x) jointly generated with A210561; see the Formula section.

1, 2, 2, 2, 5, 4, 2, 6, 12, 8, 2, 6, 17, 28, 16, 2, 6, 18, 46, 64, 32, 2, 6, 18, 53, 120, 144, 64, 2, 6, 18, 54, 152, 304, 320, 128, 2, 6, 18, 54, 161, 424, 752, 704, 256, 2, 6, 18, 54, 162, 474, 1152, 1824, 1536, 512, 2, 6, 18, 54, 162, 485, 1372, 3056, 4352

Offset: 1

Author

Clark Kimberling, Mar 22 2012

Keywords

• nonn
• tabl
• easy

Comments

Last term in row n: 2^(n-1)

Limiting row: 2*3^(n-1)

Alternating row sums: 1,0,1,0,1,0,1,0,...

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

Examples

			First five rows:
  1
  2   2
  2   5   4
  2   6   12   8
  2   6   17   28   16
First three polynomials v(n,x): 1, 2 + 2*x, 2 + 5*x + 4*x^2.
		

Links

• P. Bala, A note on the diagonals of a proper Riordan Array

Crossrefs

Row sums A005409. Cf. A208510, A210561.

Programs

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 16;
u[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + 1;
v[n_, x_] := (x + 1)*u[n - 1, x] + x*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[%]  (* A210561 *)
Table[Expand[v[n, x]], {n, 1, z}]
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]  (* A210562 *)

Formula

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

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

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

From Peter Bala, Mar 06 2017: (Start)

T(n,k) = 2*T(n-1,k-1) + T(n-2,k-1).

E.g.f. for the n-th subdiagonal: exp(2*x)*(2 + 2*x + 2*x^2/2! + 2*x^3/3! + ... + 2*x^(n-1)/(n-1)! + x^n/n!).

Riordan array ((1 + x)/(1 - x), x*(2 + x)).

Row sums A005409 (except for the initial term). (End)