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A157383 A partition product of Stirling_1 type [parameter k = -3] with biggest-part statistic (triangle read by rows).

%I A157383 #2 Mar 30 2012 17:27:11
%S A157383 1,1,3,1,9,12,1,45,48,60,1,165,480,300,360,1,855,3840,3600,2160,2520,
%T A157383 1,3843,29400,46200,30240,17640,20160,1,21819,272832,520800,443520,
%U A157383 282240,161280,181440
%N A157383 A partition product of Stirling_1 type [parameter k = -3] with biggest-part statistic (triangle read by rows).
%C A157383 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = -3,
%C A157383 summed over parts with equal biggest part (see the Luschny link).
%C A157383 Underlying partition triangle is A144353.
%C A157383 Same partition product with length statistic is A046089.
%C A157383 Diagonal a(A000217(n)) = rising_factorial(3,n-1), A001710(n+1).
%C A157383 Row sum is A049376.
%H A157383 Peter Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html"> Counting with Partitions</a>.
%H A157383 Peter Luschny, <a href="http://www.luschny.de/math/seq/stirling1partitions.html"> Generalized Stirling_1 Triangles</a>.
%F A157383 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
%F A157383 T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
%F A157383 1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
%F A157383 f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n-1).
%Y A157383 Cf. A157386, A157385, A157384, A157400, A126074, A157391, A157392, A157393, A157394, A157395
%K A157383 easy,nonn,tabl
%O A157383 1,3
%A A157383 _Peter Luschny_, Mar 07 2009, Mar 14 2009