A352399 - OEIS

%I #19 Jul 05 2022 19:36:21

%S 1,0,1,0,2,1,0,12,6,1,0,146,60,12,1,0,3060,970,180,20,1,0,101642,

%T 24180,3750,420,30,1,0,5106612,901334,110040,10990,840,42,1,0,

%U 377403266,49347228,4567976,376320,27020,1512,56,1,0,40299722580,3923052354,269812620,17322648,1071000,58716,2520,72,1

%N Triangular array read by rows: T(n,k) is the number of partial order relations on [n] that have exactly k components, n>=0, 0<=k<=n.

%F E.g.f.: A(x)^y where A(x) is the e.g.f. for A001035.

%e Triangle T(n,k) begins:

%e   1;

%e   0,    1;

%e   0,    2,   1;

%e   0,   12,   6,   1;

%e   0,  146,  60,  12,  1;

%e   0, 3060, 970, 180, 20, 1;

%e   ...

%t nn = 8; A[x_] := Total[Cases[Import["https://oeis.org/A001035/b001035.txt",

%t       "Table"], {_, _}][[All, 2]]* Table[x^(i - 1)/(i - 1)!, {i, 1, 19}]];

%t Table[Take[(Range[0, nn]! CoefficientList[Series[A[x]^y, {x, 0, nn}], {x, y}])[[i]], i], {i, 1, nn}] // Grid

%Y Cf. A001927 (column 1), A001035 (row sums), A046908.

%K nonn,tabl

%O 0,5

%A _Geoffrey Critzer_, Jul 05 2022