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A210561 Triangle of coefficients of polynomials u(n,x) jointly generated with A210562; see the Formula section. 2
1, 1, 2, 1, 3, 4, 1, 3, 8, 8, 1, 3, 9, 20, 16, 1, 3, 9, 26, 48, 32, 1, 3, 9, 27, 72, 112, 64, 1, 3, 9, 27, 80, 192, 256, 128, 1, 3, 9, 27, 81, 232, 496, 576, 256, 1, 3, 9, 27, 81, 242, 656, 1248, 1280, 512, 1, 3, 9, 27, 81, 243, 716, 1808, 3072, 2816, 1024, 1, 3, 9 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Last term in row n:  2^(n-1)

Limiting row:  3^(k-1)

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

LINKS

Table of n, a(n) for n=1..69.

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

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 n-th subdiagonal: exp(2*x)*(1 + x + x^2/2! + x^3/3! + ... + x^n/n!). Cf. A004070.

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

Row sums A048739.

(End)

EXAMPLE

First five rows:

1

1...2

1...3...4

1...3...8...8

1...3...9...20...16

First three polynomials u(n,x): 1, 1 + 2x, 1 + 3x + 4x^2.

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] + 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[%]  (* A210559 *)

Table[Expand[v[n, x]], {n, 1, z}]

cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];

TableForm[cv]

Flatten[%]  (* A210560 *)

CROSSREFS

Cf. A210562, A208510, A004070, A048739.

Sequence in context: A186975 A027422 A135086 * A210549 A187002 A177226

Adjacent sequences:  A210558 A210559 A210560 * A210562 A210563 A210564

KEYWORD

nonn,tabl,easy

AUTHOR

Clark Kimberling, Mar 22 2012

STATUS

approved

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Last modified October 22 03:04 EDT 2019. Contains 328315 sequences. (Running on oeis4.)