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A partition product of Stirling_1 type [parameter k = 4] with biggest-part statistic (triangle read by rows).
12

%I #4 Jun 23 2023 06:29:28

%S 1,1,4,1,12,12,1,72,48,24,1,280,600,120,24,1,1740,4560,1800,144,0,1,

%T 8484,40740,21000,2520,0,0,1,57232,390432,223440,33600,0,0,0,1,328752,

%U 3811248,2845584,438480,0,0,0,0,1,2389140

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

%C Partition product of prod_{j=0..n-2}(k-n+j+2) and n! at k = 4,

%C summed over parts with equal biggest part (see the Luschny link).

%C Underlying partition triangle is A144878.

%C Same partition product with length statistic is A049424.

%C Diagonal a(A000217(n)) = falling_factorial(4,n-1), row in A008279

%C Row sum is A049427.

%H Peter Luschny, <a href="http://www.luschny.de/math/seq/CountingWithPartitions.html"> Counting with Partitions</a>.

%H Peter Luschny, <a href="http://www.luschny.de/math/seq/stirling1partitions.html"> Generalized Stirling_1 Triangles</a>.

%F T(n,0) = [n = 0] (Iverson notation) and for n > 0 and 1 <= m <= n

%F T(n,m) = Sum_{a} M(a)|f^a| where a = a_1,..,a_n such that

%F 1*a_1+2*a_2+...+n*a_n = n and max{a_i} = m, M(a) = n!/(a_1!*..*a_n!),

%F f^a = (f_1/1!)^a_1*..*(f_n/n!)^a_n and f_n = product_{j=0..n-2}(j-n+6).

%e 1

%e 1 4

%e 1 12 12

%e 1 72 48 24

%e 1 280 600 120 24

%e 1 1740 4560 1800 144 0

%e 1 8484 40740 21000 2520 0 0

%e 1 57232 390432 223440 33600 0 0 0

%e 1 328752 3811248 2845584 438480 0 0 0 0

%e 1 2389140

%Y Cf. A157386, A157385, A157384, A157383, A157400, A157391, A157392, A157393, A157394, A157395

%K easy,nonn,tabl

%O 1,3

%A _Peter Luschny_, Mar 07 2009, Mar 14 2009