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A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle).

%I A144355 #10 Aug 29 2019 16:44:19
%S A144355 1,5,1,30,15,1,210,120,75,30,1,1680,1050,1500,300,375,50,1,15120,
%T A144355 10080,15750,9000,3150,9000,1875,600,1125,75,1,151200,105840,176400,
%U A144355 220500,35280,110250,63000,78750,7350,31500,13125,1050,2625,105,1,1663200,1209600,2116800
%N A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle).
%C A144355 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(5;n,k) with the k-th partition of n in A-St order.
%C A144355 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A144355 Fifth member (K=5) in the family M31(K) of partition number arrays.
%C A144355 If M31(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(5)|:= A049353.
%H A144355 W. Lang, <a href="/A144355/a144355.txt">First 10 rows of the array and more.</a>
%H A144355 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 A144355 a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(5;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)|= A001720(n+3) = (n+3)!/4!, 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 A144355 [1];[5,1];[30,15,1];[210,120,75,30,1];[1680,1050,1500,300,375,50,1];...
%e A144355 a(4,3)= 75 = 3*|S1(5;2,1)|^2. The relevant partition of 4 is (2^2).
%Y A144355 A049378 (row sums).
%Y A144355 A144354 (M31(4) array), A144356 (M31(6) array).
%K A144355 nonn,easy,tabf
%O A144355 1,2
%A A144355 _Wolfdieter Lang_ Oct 09 2008, Oct 28 2008