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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). 25
 1, 1, 2, 1, 6, 6, 1, 24, 24, 24, 1, 80, 180, 120, 120, 1, 330, 1200, 1080, 720, 720, 1, 1302, 7770, 10920, 7560, 5040, 5040, 1, 5936, 57456, 102480, 87360, 60480, 40320, 40320, 1, 26784, 438984, 970704, 1103760, 786240, 544320, 362880, 362880 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS 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). It shares this property with the signless Lah numbers. Underlying partition triangle is A130561. Same partition product with length statistic is A105278. Diagonal a(A000217) = A000142. Row sum is A000262. 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 LINKS Table of n, a(n) for n=1..45. Peter Luschny, Counting with Partitions. Peter Luschny, Generalized Stirling_1 Triangles. Peter Luschny, Generalized Stirling_2 Triangles. FORMULA T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n. T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,...,a_n such that 1*a_1 + 2*a_2 + ... + n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!), f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = Product_{j=0..n-1} (-j-1) OR f_n = Product_{j=0..n-2} (j-n) since both have the same absolute value n!. 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 EXAMPLE Triangle starts: 1; 1, 2; 1, 6, 6; 1, 24, 24, 24; 1, 80, 180, 120, 120; 1, 330, 1200, 1080, 720, 720; ... MAPLE egf:= k-> exp((x^(k+1)-x)/(x-1))-exp((x^k-x)/(x-1)): T:= (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n): seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Oct 10 2015 MATHEMATICA 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 *) CROSSREFS Cf. A157396, A157397, A157398, A157399, A080510, A157401, A157402, A157403, A157404, A157405, A157386, A157385, A157384, A157383, A126074, A157391, A157392, A157393, A157394, A157395. Sequence in context: A110098 A244888 A130561 * A091599 A048999 A066667 Adjacent sequences: A157397 A157398 A157399 * A157401 A157402 A157403 KEYWORD easy,nonn,tabl AUTHOR Peter Luschny, Mar 09 2009, Mar 14 2009 STATUS approved

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Last modified April 25 09:38 EDT 2024. Contains 371967 sequences. (Running on oeis4.)