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A145363 Partition number array, called M31hat(-2).

%I A145363 #14 May 27 2025 10:30:46
%S A145363 1,2,1,2,2,1,0,2,4,2,1,0,0,4,2,4,2,1,0,0,0,4,0,4,8,2,4,2,1,0,0,0,0,0,
%T A145363 0,4,8,0,4,8,2,4,2,1,0,0,0,0,0,0,0,0,0,8,0,0,4,8,16,0,4,8,2,4,2,1,0,0,
%U A145363 0,0,0,0,0,0,0,0,0,8,0,0,0,0,8,16,0,0,4,8,16,0,4,8,2,4,2,1,0,0,0,0,0,0,0,0
%N A145363 Partition number array, called M31hat(-2).
%C A145363 If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2 or 3 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
%C A145363 Second member (K=2) in the family M31hat(-K) of partition number arrays.
%C A145363 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A145363 This array is array A144358 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144358/A036040'. E.g. a(4,3)= 4 = 12/3 = A144358(4,3)/A036040(4,3).
%C A145363 If M31hat(-2;n,k) is summed over those k belonging to partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-2):= A145364.
%H A145363 Wolfdieter Lang, <a href="/A145363/a145363.txt">First 10 rows of the array and more.</a>
%H A145363 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 A145363 a(n,k) = product(S1(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1) = A008279(2,n-1) = [1,2,1,0,0,0,...], 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 A145363 Triangle begins
%e A145363   [1];
%e A145363   [2,1];
%e A145363   [2,2,1];
%e A145363   [0,2,4,2,1];
%e A145363   [0,0,4,2,4,2,1];
%e A145363   ...
%e A145363 a(4,3)= 4 = S1(-2;2,1)^2. The relevant partition of 4 is (2^2).
%Y A145363 Cf. A145361 (M31hat(-1)). A145366 (M31hat(-3)).
%K A145363 nonn,easy,tabf
%O A145363 1,2
%A A145363 _Wolfdieter Lang_, Oct 17 2008