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A157400 A partition product with biggest-part statistic of Stirling_1 type (with parameter k = -2) as well as of Stirling_2 type (with parameter k = -2), (triangle read by rows).

%I A157400 #22 Nov 22 2021 02:29:10
%S A157400 1,1,2,1,6,6,1,24,24,24,1,80,180,120,120,1,330,1200,1080,720,720,1,
%T A157400 1302,7770,10920,7560,5040,5040,1,5936,57456,102480,87360,60480,40320,
%U A157400 40320,1,26784,438984,970704,1103760,786240,544320,362880,362880
%N A157400 A partition product with biggest-part statistic of Stirling_1 type (with parameter k = -2) as well as of Stirling_2 type (with parameter k = -2), (triangle read by rows).
%C A157400 Partition product of Product_{j=0..n-1} ((k+1)*j - 1) and n! at k = -2, summed over parts with equal biggest part (Stirling_2 type) as well as partition product of Product_{j=0..n-2} (k-n+j+2) and n! at k = -2 (Stirling_1 type).
%C A157400 It shares this property with the signless Lah numbers.
%C A157400 Underlying partition triangle is A130561.
%C A157400 Same partition product with length statistic is A105278.
%C A157400 Diagonal a(A000217) = A000142.
%C A157400 Row sum is A000262.
%C A157400 T(n,k) is the number of nilpotent elements in the symmetric inverse semigroup (partial bijections) on [n] having index k. Equivalently, T(n,k) is the number of directed acyclic graphs on n labeled nodes with every node having indegree and outdegree at most one and the longest path containing exactly k nodes. - _Geoffrey Critzer_, Nov 21 2021
%H A157400 Peter Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html"> Counting with Partitions</a>.
%H A157400 Peter Luschny, <a href="http://www.luschny.de/math/seq/stirling1partitions.html"> Generalized Stirling_1 Triangles</a>.
%H A157400 Peter Luschny, <a href="http://www.luschny.de/math/seq/stirling2partitions.html"> Generalized Stirling_2 Triangles</a>.
%F A157400 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n.
%F A157400 T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,...,a_n such that
%F A157400 1*a_1 + 2*a_2 + ... + n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
%F A157400 f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = Product_{j=0..n-1} (-j-1)
%F A157400 OR f_n = Product_{j=0..n-2} (j-n) since both have the same absolute value n!.
%F A157400 E.g.f. of column k: exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)). - _Alois P. Heinz_, Oct 10 2015
%e A157400 Triangle starts:
%e A157400   1;
%e A157400   1,   2;
%e A157400   1,   6,    6;
%e A157400   1,  24,   24,   24;
%e A157400   1,  80,  180,  120, 120;
%e A157400   1, 330, 1200, 1080, 720, 720;
%e A157400   ...
%p A157400 egf:= k-> exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)):
%p A157400 T:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
%p A157400 seq(seq(T(n, k), k=1..n), n=1..10);  # _Alois P. Heinz_, Oct 10 2015
%t A157400 egf[k_] := Exp[(x^(k+1)-x)/(x-1)] - Exp[(x^k-x)/(x-1)]; T[n_, k_] := n! * SeriesCoefficient[egf[k], {x, 0, n}]; Table[Table[T[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* _Jean-François Alcover_, Oct 11 2015, after _Alois P. Heinz_ *)
%Y A157400 Cf. A157396, A157397, A157398, A157399, A080510, A157401, A157402, A157403, A157404, A157405, A157386, A157385, A157384, A157383, A126074, A157391, A157392, A157393, A157394, A157395.
%K A157400 easy,nonn,tabl
%O A157400 1,3
%A A157400 _Peter Luschny_, Mar 09 2009, Mar 14 2009