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A145366 Partition number array, called M31hat(-3).

%I A145366 #13 May 27 2025 10:30:38
%S A145366 1,3,1,6,3,1,6,6,9,3,1,0,6,18,6,9,3,1,0,0,18,36,6,18,27,6,9,3,1,0,0,0,
%T A145366 36,0,18,36,54,6,18,27,6,9,3,1,0,0,0,0,36,0,0,36,54,108,0,18,36,54,81,
%U A145366 6,18,27,6,9,3,1,0,0,0,0,0,0,0,0,36,0,108,216,0,0,36,54,108,162,0,18,36,54,81
%N A145366 Partition number array, called M31hat(-3).
%C A145366 If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2,3 or 4 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
%C A145366 Third member (K=3) in the family M31hat(-K) of partition number arrays.
%C A145366 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A145366 This array is array A144877 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144877/A036040'. E.g. a(4,3)= 9 = 27/3 = A144877(4,3)/A036040(4,3).
%C A145366 If M31hat(-3;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-3):= A145367.
%H A145366 Wolfdieter Lang, <a href="/A145366/a145366.txt">First 10 rows of the array and more.</a>
%H A145366 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 A145366 a(n,k) = product(S1(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1) = A008279(3,n-1) = [1,3,6,6,0,0,0,...], 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 A145366 Triangle begins:
%e A145366   [1];
%e A145366   [3,1];
%e A145366   [6,3,1];
%e A145366   [6,6,9,3,1];
%e A145366   [0,6,18,6,9,3,1];
%e A145366   ...
%e A145366 a(4,3)= 9 = S1(-3;2,1)^2. The relevant partition of 4 is (2^2).
%Y A145366 Cf. A145363 (M31hat(-2)). A145369 (M31hat(-4)).
%K A145366 nonn,easy,tabf
%O A145366 1,2
%A A145366 _Wolfdieter Lang_, Oct 17 2008