A236376 Riordan array ((1-x+x^2)/(1-x)^2, x/(1-x)^2).

1, 1, 1, 2, 3, 1, 3, 7, 5, 1, 4, 14, 16, 7, 1, 5, 25, 41, 29, 9, 1, 6, 41, 91, 92, 46, 11, 1, 7, 63, 182, 246, 175, 67, 13, 1, 8, 92, 336, 582, 550, 298, 92, 15, 1, 9, 129, 582, 1254, 1507, 1079, 469, 121, 17, 1, 10, 175, 957, 2508, 3718, 3367, 1925, 696, 154

Offset

0,4

Comments

Triangle T(n,k), read by rows, given by (1, 1, -1, 1, 0, 0, 0, 0, 0, 0, 0, ...) DELTA (1, 0, 0, 0, 0, 0, 0, 0, 0, 0, ...) where DELTA is the operator defined in A084938.

Row sums are A111282(n+1) = A025169(n-1).

Diagonal sums are A122391(n+1) = A003945(n-1).

Links

Table of n, a(n) for n=0..63.

Formula

G.f.: (1 - x + x^2)/(1 - 2*x - x*y + x^2).

T(n,k) = 2*T(n-1,k) + T(n-1,k-1) - T(n-2,k), T(0,0) = T(1,0) = T(1,1) = 1, T(2,0) = 2, T(2,1) = 3, T(2,2) = 1, T(n,k) = 0 if k < 0 or k > n.

The Riordan square (see A321620) of 1 + x/(1 - x)^2. - Peter Luschny, Mar 06 2022

Example

Triangle begins:
  1;
  1,  1;
  2,  3,   1;
  3,  7,   5,   1;
  4, 14,  16,   7,   1;
  5, 25,  41,  29,   9,  1;
  6, 41,  91,  92,  46, 11,  1;
  7, 63, 182, 246, 175, 67, 13, 1;

Maple

# The function RiordanSquare is defined in A321620.
RiordanSquare(1+x/(1-x)^2, 8); # Peter Luschny, Mar 06 2022

Mathematica

CoefficientList[#, y] & /@
CoefficientList[
Series[(1 - x + x^2)/(1 - 2*x - x*y + x^2), {x, 0, 12}], x] (* Wouter Meeussen, Jan 25 2014 *)

Crossrefs

Cf. Columns: A028310, A004006.

Cf. Diagonals: A000012, A005408, A130883.

Cf. Similar sequences: A078812, A085478, A111125, A128908, A165253, A207606.

Cf. A321620.

Sequence in context: A343627 A188107 A174014 * A063967 A059397 A209567

Adjacent sequences: A236373 A236374 A236375 * A236377 A236378 A236379

Keyword

nonn,tabl

Author

Philippe Deléham, Jan 24 2014

Status

approved