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
KEYWORD
AUTHOR
Christian G. Bower, Jan 19 2004
STATUS
approved