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

%I A259454 #19 Feb 18 2016 12:18:42
%S A259454 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,
%T A259454 571,374,79,1,21,172,700,1496,1632,809,133,1,24,232,1164,3261,5116,
%U A259454 4270,1668,221,1,27,301,1799,6271,13013,15754,10446,3316,364
%N A259454 Triangle T(n,k) (0 <= k <= n) read by rows, arising from the study of rook polynomials.
%C A259454 See Riordan 1954 page 18 equation (9). - _Michael Somos_, Aug 26 2015
%H A259454 Alois P. Heinz, <a href="/A259454/b259454.txt">Rows n = 0..140, flattened</a>
%H A259454 J. Riordan, <a href="/A000211/a000211.pdf">Discordant permutations</a>, Scripta Math., 20 (1954), 14-23. [Annotated scanned copy] (See triangle on page 18)
%F A259454 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.
%F A259454 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
%e A259454 Triangle T(n,k) begins:
%e A259454 1;
%e A259454 1,  3;
%e A259454 1,  6,  7;
%e A259454 1,  9,  22,   14;
%e A259454 1, 12,  46,   64,   26;
%e A259454 1, 15,  79,  177,  162,   46;
%e A259454 1, 18, 121,  380,  571,  374,   79;
%e A259454 1, 21, 172,  700, 1496, 1632,  809,  133;
%e A259454 1, 24, 232, 1164, 3261, 5116, 4270, 1668, 221;
%e A259454 G.f. = 1 + (1 + 3*t)*u + (1 + 6*t + 7*t^2)*u^2 + ...
%p A259454 T:= proc(n, k) option remember; `if`(k<0 or k>n, 0,
%p A259454       T(n-1, k) +2*T(n-1, k-1) +T(n-2, k-1)
%p A259454      -T(n-3, k-3) +`if`(n=k, 1, 0))
%p A259454     end:
%p A259454 seq(seq(T(n, k), k=0..n), n=0..10);  # _Alois P. Heinz_, Jul 02 2015
%t A259454 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 *)
%o A259454 (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 */
%Y A259454 Some diagonals: A001924, A001925, A001926.
%K A259454 nonn,tabl
%O A259454 0,3
%A A259454 _N. J. A. Sloane_, Jun 28 2015
%E A259454 More terms from _Alois P. Heinz_, Jul 02 2015