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A157394 A partition product of Stirling_1 type [parameter k = 4] with biggest-part statistic (triangle read by rows).

%I A157394 #4 Jun 23 2023 06:29:28
%S A157394 1,1,4,1,12,12,1,72,48,24,1,280,600,120,24,1,1740,4560,1800,144,0,1,
%T A157394 8484,40740,21000,2520,0,0,1,57232,390432,223440,33600,0,0,0,1,328752,
%U A157394 3811248,2845584,438480,0,0,0,0,1,2389140
%N A157394 A partition product of Stirling_1 type [parameter k = 4] with biggest-part statistic (triangle read by rows).
%C A157394 Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 4,
%C A157394 summed over parts with equal biggest part (see the Luschny link).
%C A157394 Underlying partition triangle is A144878.
%C A157394 Same partition product with length statistic is A049424.
%C A157394 Diagonal a(A000217(n)) = falling_factorial(4,n-1), row in A008279
%C A157394 Row sum is A049427.
%H A157394 Peter Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html"> Counting with Partitions</a>.
%H A157394 Peter Luschny, <a href="http://www.luschny.de/math/seq/stirling1partitions.html"> Generalized Stirling_1 Triangles</a>.
%F A157394 T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n
%F A157394 T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that
%F A157394 1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),
%F A157394 f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+6).
%e A157394 1
%e A157394 1       4
%e A157394 1      12       12
%e A157394 1      72       48       24
%e A157394 1     280      600      120      24
%e A157394 1    1740     4560     1800     144  0
%e A157394 1    8484    40740    21000    2520  0  0
%e A157394 1   57232   390432   223440   33600  0  0  0
%e A157394 1  328752  3811248  2845584  438480  0  0  0  0
%e A157394 1  2389140
%Y A157394 Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395
%K A157394 easy,nonn,tabl
%O A157394 1,3
%A A157394 _Peter Luschny_, Mar 07 2009, Mar 14 2009