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A145372 Partition number array, called M31hat(-5).

%I A145372 #13 May 27 2025 10:35:21
%S A145372 1,5,1,20,5,1,60,20,25,5,1,120,60,100,20,25,5,1,120,120,300,400,60,
%T A145372 100,125,20,25,5,1,0,120,600,1200,120,300,400,500,60,100,125,20,25,5,
%U A145372 1,0,0,600,2400,3600,120,600,1200,1500,2000,120,300,400,500,625,60,100,125,20
%N A145372 Partition number array, called M31hat(-5).
%C A145372 If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2,3,4,5 or 6 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
%C A145372 Fifth member (K=5) in the family M31hat(-K) of partition number arrays.
%C A145372 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A145372 This array is array A144879 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144879/A036040'. E.g. a(4,3)= 25 = 75/3 = A144879(4,3)/A036040(4,3).
%C A145372 If M31hat(-5;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-5):= A145373.
%H A145372 Wolfdieter Lang, <a href="/A145372/a145372.txt">First 10 rows of the array and more.</a>
%H A145372 Wolfdieter 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 A145372 a(n,k) = product(S1(-5;j,1)^e(n,k,j),j=1..n) with S1(-5;n,1) = A008279(5,n-1) = [1,5,20,60,120,120,0,0,0,...], 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.
%e A145372 Triangle begins;
%e A145372   [1];
%e A145372   [5,1];
%e A145372   [20,5,1];
%e A145372   [60,20,25,5,1];
%e A145372   [120,60,100,20,25,5,1];
%e A145372   ...
%e A145372 a(4,3)= 25 = S1(-4;2,1)^2. The relevant partition of 4 is (2^2).
%Y A145372 Cf. A145369 (M31hat(-4)).
%K A145372 nonn,easy,tabf
%O A145372 1,2
%A A145372 _Wolfdieter Lang_, Oct 17 2008