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A355005 Table read by rows. T(n, k) = n*((k + n)!)^2/((k + n)*(n!)^2*k!) for n > 0 and T(0, 0) = 1.

%I A355005 #6 Jun 15 2022 11:52:52
%S A355005 1,1,2,1,6,36,1,12,120,1200,1,20,300,4200,58800,1,30,630,11760,211680,
%T A355005 3810240,1,42,1176,28224,635040,13970880,307359360,1,56,2016,60480,
%U A355005 1663200,43908480,1141620480,29682132480,1,72,3240,118800,3920400,122316480,3710266560,111307996800,3339239904000
%N A355005 Table read by rows. T(n, k) = n*((k + n)!)^2/((k + n)*(n!)^2*k!) for n > 0 and T(0, 0) = 1.
%F A355005 T(n, k) = Lah(k + n, n), where Lah denotes the unsigned Lah numbers A271703.
%e A355005 [0] 1;
%e A355005 [1] 1,  2;
%e A355005 [2] 1,  6,   36;
%e A355005 [3] 1, 12,  120,  1200;
%e A355005 [4] 1, 20,  300,  4200,   58800;
%e A355005 [5] 1, 30,  630, 11760,  211680,  3810240;
%e A355005 [6] 1, 42, 1176, 28224,  635040, 13970880,  307359360;
%e A355005 [7] 1, 56, 2016, 60480, 1663200, 43908480, 1141620480, 29682132480;
%p A355005 T := (n, k) -> ifelse(n = 0, 1, n*((k + n)!)^2 / ((k + n)*(n!)^2*k!)):
%p A355005 seq(seq(T(n, k), k = 0..n), n = 0..8);
%Y A355005 T(n, 1) = A002378, T(n, n) = A187535, A355004 (row sums), A271703 (Lah).
%K A355005 nonn,tabl
%O A355005 0,3
%A A355005 _Peter Luschny_, Jun 15 2022