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A259454 Triangle T(n,k) (0 <= k <= n) read by rows, arising from the study of rook polynomials. 1
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 (list; table; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A210749 A199662

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

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Last modified November 18 07:22 EST 2019. Contains 329252 sequences. (Running on oeis4.)