cp's OEIS Frontend

A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.

%I A351884 #19 Mar 01 2022 12:04:58
%S A351884 1,0,1,0,2,0,6,6,0,24,24,0,120,240,0,720,1440,720,0,5040,15120,5040,0,
%T A351884 40320,120960,80640,0,362880,1451520,1088640,0,3628800,14515200,
%U A351884 14515200,3628800,0,39916800,199584000,199584000,39916800,0,479001600,2395008000,3353011200,958003200
%N A351884 Irregular triangle read by rows: T(n,k) is the number of sets of lists with distinct block sizes (as in A088311(n)) and containing exactly k lists.
%F A351884 E.g.f.: Product_{i>=1} (1 + y*x^i).
%F A351884 Sum_{k=0..A003056(n)} (-1)^k * T(n,k) = A293140(n). - _Alois P. Heinz_, Feb 23 2022
%e A351884 Triangle T(n,k) begins:
%e A351884   1;
%e A351884   0,     1;
%e A351884   0,     2;
%e A351884   0,     6,      6;
%e A351884   0,    24,     24;
%e A351884   0,   120,    240;
%e A351884   0,   720,   1440,   720;
%e A351884   0,  5040,  15120,  5040;
%e A351884   0, 40320, 120960, 80640;
%e A351884   ...
%p A351884 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A351884       b(n, i-1)+expand(x*b(n-i, min(i-1, n-i)))*n!/(n-i)!))
%p A351884     end:
%p A351884 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A351884 seq(T(n), n=0..12);  # _Alois P. Heinz_, Feb 23 2022
%t A351884 nn = 13; Prepend[Map[Prepend[#, 0] &, Drop[Map[Select[#, # > 0 &] &,Range[0, nn]! CoefficientList[Series[Product[1 + y x^i, {i, 1, nn}], {x, 0, nn}],{x,y}]], 1]], {1}] // Grid
%Y A351884 Columns k=0-1 give: A000007, A000142 (for n>=1).
%Y A351884 Cf. A088311 (row sums).
%Y A351884 T(A000217(n),n) gives A052295.
%Y A351884 Cf. A003056, A293140.
%K A351884 nonn,tabf
%O A351884 0,5
%A A351884 _Geoffrey Critzer_, Feb 23 2022