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A157403 A partition product of Stirling_2 type [parameter k = 3] with biggest-part statistic (triangle read by rows).

%I A157403 #2 Mar 30 2012 17:27:11
%S A157403 1,1,3,1,9,21,1,45,84,231,1,165,840,1155,3465,1,855,8610,13860,20790,
%T A157403 65835,1,3843,64680,250635,291060,460845,1514205,1,21819,689136,
%U A157403 3969735,6015240,7373520,12113640,40883535,1,114075
%N A157403 A partition product of Stirling_2 type [parameter k = 3] with biggest-part statistic (triangle read by rows).
%C A157403 Partition product of prod_{j=0..n-1}((k + 1)*j - 1) and n! at k = 3,
%C A157403 summed over parts with equal biggest part (see the Luschny link).
%C A157403 Underlying partition triangle is A143173.
%C A157403 Same partition product with length statistic is A000369.
%C A157403 Diagonal a(A000217) = A008545
%C A157403 Row sum is A016036.
%H A157403 Peter Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html"> Counting with Partitions</a>.
%H A157403 Peter Luschny, <a href="http://www.luschny.de/math/seq/stirling2partitions.html"> Generalized Stirling_2 Triangles</a>.
%F A157403 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
%F A157403 T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
%F A157403 1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
%F A157403 f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-1}(4*j - 1).
%Y A157403 Cf. A157396, A157397, A157398, A157399, A157400, A080510, A157401, A157402, A157404, A157405
%K A157403 easy,nonn,tabl
%O A157403 1,3
%A A157403 _Peter Luschny_, Mar 09 2009