A259454 Triangle T(n,k) (0 <= k <= n) read by rows, arising from the study of rook polynomials.

1, 1, 3, 1, 6, 7, 1, 9, 22, 14, 1, 12, 46, 64, 26, 1, 15, 79, 177, 162, 46, 1, 18, 121, 380, 571, 374, 79, 1, 21, 172, 700, 1496, 1632, 809, 133, 1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221, 1, 27, 301, 1799, 6271, 13013, 15754, 10446, 3316, 364

Offset

0,3

Comments

See Riordan 1954 page 18 equation (9). - Michael Somos, Aug 26 2015

Links

Alois P. Heinz, Rows n = 0..140, flattened

J. Riordan, Discordant permutations, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] (See triangle on page 18)

Formula

From Eq. (11) of Riordan (1954): T(n,k) = T(n-1,k) + 2*T(n-1,k-1) + T(n-2,k-1) - T(n-3,k-3) + delta(n,k), where delta(n,k)=1 iff n=k, otherwise 0.

Sum_{n, k} T(n, k) * x^n*y^k = 1 / ((1 - y*x) * (1 - (1 + 2*y)*x - y*x^2 + y^3*x^3)). - Michael Somos, Aug 26 2015

Example

Triangle T(n,k) begins:
1;
1,  3;
1,  6,  7;
1,  9,  22,   14;
1, 12,  46,   64,   26;
1, 15,  79,  177,  162,   46;
1, 18, 121,  380,  571,  374,   79;
1, 21, 172,  700, 1496, 1632,  809,  133;
1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221;
G.f. = 1 + (1 + 3*t)*u + (1 + 6*t + 7*t^2)*u^2 + ...

Maple

T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
      T(n-1, k) +2*T(n-1, k-1) +T(n-2, k-1)
     -T(n-3, k-3) +`if`(n=k, 1, 0))
    end:
seq(seq(T(n, k), k=0..n), n=0..10);  # Alois P. Heinz, Jul 02 2015

Mathematica

T[n_, k_] /; 0 <= k <= n := T[n, k] = T[n-1, k] + 2*T[n-1, k-1] + T[n-2, k - 1] - T[n-3, k-3] + Boole[n == k]; T[_, _] = 0; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Jean-François Alcover, Feb 18 2016 *)

Prog

(PARI) {T(n, k) = polcoeff( polcoeff( 1 / ((1 - y*x) * (1 - (1 + 2*y)*x - y*x^2 + y^3*x^3)) + x * O(x^n), n), k)}; /* Michael Somos, Aug 26 2015 */

Crossrefs

Some diagonals: A001924, A001925, A001926.

Sequence in context: A239385 A124929 A208766 * A209696 A338995 A359574

Adjacent sequences: A259451 A259452 A259453 * A259455 A259456 A259457

Keyword

nonn,tabl

Author

N. J. A. Sloane, Jun 28 2015

Extensions

More terms from Alois P. Heinz, Jul 02 2015

Status

approved