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A144356 Partition number array, called M31(6), related to A049374(n,m)= |S1(6;n,m)| (generalized Stirling triangle).

%I A144356 #10 Aug 29 2019 17:53:13
%S A144356 1,6,1,42,18,1,336,168,108,36,1,3024,1680,2520,420,540,60,1,30240,
%T A144356 18144,30240,17640,5040,15120,3240,840,1620,90,1,332640,211680,381024,
%U A144356 493920,63504,211680,123480,158760,11760,52920,22680,1470,3780,126,1,3991680,2661120
%N A144356 Partition number array, called M31(6), related to A049374(n,m)= |S1(6;n,m)| (generalized Stirling triangle).
%C A144356 Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(6;n,k) with the k-th partition of n in A-St order.
%C A144356 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A144356 Sixth member (K=6) in the family M31(K) of partition number arrays.
%C A144356 If M31(6;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(6)|:= A049374.
%H A144356 W. Lang, <a href="/A144356/a144356.txt">First 10 rows of the array and more.</a>
%H A144356 W. Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL12/Lang/lang.html">Combinatorial Interpretation of Generalized Stirling Numbers</a>, J. Int. Seqs. Vol. 12 (2009) 09.3.3.
%F A144356 a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(6;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(6;j,1)|^e(n,k,j),j=1..n) with |S1(6;n,1)|= A001725(n+4) = (n+4)!/5!, n>=1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. M3(n,k)=A036040.
%e A144356 [1];[6,1];[42,18,1];[336,168,108,36,1];[3024,1680,2520,420,540,60,1];...
%e A144356 a(4,3)= 108 = 3*|S1(6;2,1)|^2. The relevant partition of 4 is (2^2).
%Y A144356 A049402 (row sums).
%Y A144356 A144355 (M31(5) array).
%K A144356 nonn,easy,tabf
%O A144356 1,2
%A A144356 _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008