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

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

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.

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

Peter Bala, Notes on generalized Riordan arrays

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)

From Peter Bala, Nov 12 2025: (Start)

T(n, k) = Sum_{j = 0..n-k} 2^(2*k-n+j)*binomial(k, n-k-j).

As a square array (read by anti-diagonals) equals U * transpose(P^2), where P denotes Pascal's triangle A007318 and U denotes the Riordan array (1/(1 - x), x), that is, the lower triangular array with 1's in every position on or below the main diagonal.

The k-th column of the square array has the g.f. 1/(1 - x) * (2 + x)^k. (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 + 2*x, 1 + 3*x + 4*x^2.
From Peter Bala, Nov 12 2025: (Start)
As a square array read by anti-diagonals
  1  2  4   8  16   32   64   128 ...
  1  3  8  20  48  112  256   576 ...
  1  3  9  26  72  192  496  1248 ...
  1  3  9  27  80  232  656  1808 ...
  1  3  9  27  81  242  716  2088 ...
  1  3  9  27  81  243  728  2172 ...
  1  3  9  27  81  243  729  2186 ...
  1  3  9  27  81  243  729  2187 ...
  ...
Factorization of the square array as U * transpose(P^2):
  / 1           \ /1  2  4   8  ...    \     / 1  2  4   8  ...  \
  | 1  1        | |0  1  4  12  ...    |     | 1  3  8  20  ...  |
  | 1  1  1     | |0  0  1   6  ...    |  =  | 1  3  9  26  ...  |
  | 1  1  1  1  | |0  0  0   1  ...    |     | 1  3  9  27  ...  |
  | ...         | |...                 |     | ...               |
(End)

Maple

seq( print( seq(add(2^(2*k-n+j)*binomial(k, n-k-j), j = 0..n-k), k = 0..n) ), n = 0..10); # - Peter Bala, Nov 21 2025

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. A048739, A007318, A210562, A208510, A004070.

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