A091533 Triangle read by rows, related to Pascal's triangle.

1, 1, 1, 2, 3, 2, 3, 7, 7, 3, 5, 15, 21, 15, 5, 8, 30, 53, 53, 30, 8, 13, 58, 124, 157, 124, 58, 13, 21, 109, 273, 417, 417, 273, 109, 21, 34, 201, 577, 1029, 1239, 1029, 577, 201, 34, 55, 365, 1181, 2405, 3375, 3375, 2405, 1181, 365, 55, 89, 655, 2358, 5393, 8625, 10047, 8625, 5393, 2358, 655, 89

Offset

0,4

Links

Seiichi Manyama, Rows n = 0..139, flattened

Formula

T(n, k) = T(n-1, k) + T(n-1, k-1) + T(n-2, k) + T(n-2, k-1) + T(n-2, k-2) for n >= 2, k >= 0, with initial conditions specified by first two rows.

G.f.: A(x, y) = 1/(1-x-x*y-x^2-x^2*y-x^2*y^2).

Sum_{k = 0..n} T(n,k)*x^k = A000045(n+1), A015518(n+1), A015524(n+1), A200069(n+1) for x = 0, 1, 2, 3 respectively. - Philippe Deléham, Oct 30 2013

Sum_{k = 0..floor(n/2)} T(n-k,k) = (-1)^n*A079926(n). - Philippe Deléham, Oct 30 2013

Example

This triangle begins:
   1;
   1,   1;
   2,   3,    2;
   3,   7,    7,    3;
   5,  15,   21,   15,    5;
   8,  30,   53,   53,   30,     8;
  13,  58,  124,  157,  124,    58,   13;
  21, 109,  273,  417,  417,   273,  109,   21;
  34, 201,  577, 1029, 1239,  1029,  577,  201,   34;
  55, 365, 1181, 2405, 3375,  3375, 2405, 1181,  365,  55;
  89, 655, 2358, 5393, 8625, 10047, 8625, 5393, 2358, 655, 89;
  ...

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
     `if`(n<1, 1, add(add(T(n-i, k-j), j=0..i), i=1..2)))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jan 14 2022

Mathematica

A091533[-2, n2_] = 0; A091533[n1_, -2] = 0; A091533[-1, n2_] = 0; A091533[n1_, -1] = 0; A091533[0, 0] = 1; A091533[n1_, n2_] := A091533[n1, n2] = A091533[n1 - 1, n2] + A091533[n1, n2 - 1] + A091533[n1 - 1, n2 - 1] + A091533[n1 - 2, n2] + A091533[n1, n2 - 2]; Table[A091533[x - y, y], {x, 0, 9}, {y, 0, x}] // Flatten (* Robert P. P. McKone, Jan 14 2022 *)

Crossrefs

Row sums: A015518(n+1). Columns 0-1: A000045(n+1), A023610(n-1).

Cf. A090174, A212338 (column 2), A192364 (central terms).

Sequence in context: A085204 A228527 A055375 * A055376 A085215 A076731

Adjacent sequences: A091530 A091531 A091532 * A091534 A091535 A091536

Keyword

nonn,easy,tabl

Author

Christian G. Bower, Jan 19 2004

Status

approved