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%I #19 Mar 01 2022 12:04:58
%S 1,0,1,0,2,0,6,6,0,24,24,0,120,240,0,720,1440,720,0,5040,15120,5040,0,
%T 40320,120960,80640,0,362880,1451520,1088640,0,3628800,14515200,
%U 14515200,3628800,0,39916800,199584000,199584000,39916800,0,479001600,2395008000,3353011200,958003200
%N 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 E.g.f.: Product_{i>=1} (1 + y*x^i).
%F Sum_{k=0..A003056(n)} (-1)^k * T(n,k) = A293140(n). - _Alois P. Heinz_, Feb 23 2022
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 2;
%e 0, 6, 6;
%e 0, 24, 24;
%e 0, 120, 240;
%e 0, 720, 1440, 720;
%e 0, 5040, 15120, 5040;
%e 0, 40320, 120960, 80640;
%e ...
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1)+expand(x*b(n-i, min(i-1, n-i)))*n!/(n-i)!))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p seq(T(n), n=0..12); # _Alois P. Heinz_, Feb 23 2022
%t 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 Columns k=0-1 give: A000007, A000142 (for n>=1).
%Y Cf. A088311 (row sums).
%Y T(A000217(n),n) gives A052295.
%Y Cf. A003056, A293140.
%K nonn,tabf
%O 0,5
%A _Geoffrey Critzer_, Feb 23 2022
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