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A157393 A partition product of Stirling_1 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

%I A157393 #5 Dec 14 2013 02:15:06
%S A157393 1,1,3,1,9,6,1,45,24,6,1,165,240,30,0,1,855,1560,360,0,0,1,3843,12180,
%T A157393 3360,0,0,0,1,21819,96096,30660,0,0,0,0,1,114075,794304,318276,0,0,0,
%U A157393 0,0,1,703215,6850080,3270960,0,0,0,0,0,0,1,4125495,62516520,35053920,0,0
%N A157393 A partition product of Stirling_1 type [parameter k = 3] with biggest-part statistic (triangle read by rows).
%C A157393 Partition product prod_{j=0..n-2}(k-n+j+2) and n! at k = 3, summed over parts with equal biggest part (see the Luschny link).
%C A157393 Underlying partition triangle is A144877.
%C A157393 Same partition product with length statistic is A049410.
%C A157393 Diagonal a(A000217(n)) = falling_factorial(3,n-1), row in A008279.
%C A157393 Row sum is A049426.
%H A157393 Peter Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html"> Counting with Partitions</a>.
%H A157393 Peter Luschny, <a href="http://www.luschny.de/math/seq/stirling1partitions.html"> Generalized Stirling_1 Triangles</a>.
%F A157393 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
%F A157393 T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
%F A157393 1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
%F A157393 f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+5).
%Y A157393 Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395.
%K A157393 easy,nonn,tabl
%O A157393 1,3
%A A157393 _Peter Luschny_, Mar 07 2009, Mar 14 2009