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

1,2

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.

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.

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 + 2x , 2 + 5x + 4x^2.

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 *)

Cf. A210561, A208510.

Links

Table of n, a(n) for n=1..64.

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

Formula

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)

Crossrefs

Row sums A005409. Cf. A208510, A210561.

Sequence in context: A194684 A076737 A246119 * A208512 A208908 A209558

Adjacent sequences: A210559 A210560 A210561 * A210563 A210564 A210565

Keyword

nonn,tabl,easy

Author

Clark Kimberling, Mar 22 2012

Status

approved