cp's OEIS Frontend

A145369 Partition number array, called M31hat(-4).

%I A145369 #14 May 27 2025 10:30:31
%S A145369 1,4,1,12,4,1,24,12,16,4,1,24,24,48,12,16,4,1,0,24,96,144,24,48,64,12,
%T A145369 16,4,1,0,0,96,288,24,96,144,192,24,48,64,12,16,4,1,0,0,0,288,576,0,
%U A145369 96,288,384,576,24,96,144,192,256,24,48,64,12,16,4,1,0,0,0,0,576,0,0,288,576,384
%N A145369 Partition number array, called M31hat(-4).
%C A145369 If all positive numbers are replaced by 1 this becomes the characteristic partition array for partitions with parts 1,2,3,4 or 5 only, provided the partitions of n are ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278).
%C A145369 Fourth member (K=4) in the family M31hat(-K) of partition number arrays.
%C A145369 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A145369 This array is array A144878 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144878/A036040'. E.g. a(4,3)= 16 = 48/3 = A144878(4,3)/A036040(4,3).
%C A145369 If M31hat(-4;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-4):= A145370.
%H A145369 Wolfdieter Lang, <a href="/A145369/a145369.txt">First 10 rows of the array and more.</a>
%H A145369 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 A145369 a(n,k) = product(S1(-4;j,1)^e(n,k,j),j=1..n) with S1(-4;n,1) = A008279(4,n-1) = [1,4,12,24,24,0,0,0,...], 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 A145369 Triangle begins:
%e A145369   [1];
%e A145369   [4,1];
%e A145369   [12,4,1];
%e A145369   [24,12,16,4,1];
%e A145369   [24,24,48,12,16,4,1];
%e A145369   ...
%e A145369 a(4,3)= 16 = S1(-4;2,1)^2. The relevant partition of 4 is (2^2).
%Y A145369 Cf. A145366 (M31hat(-3)), A145372 (M31hat(-5)).
%K A145369 nonn,easy,tabf
%O A145369 1,2
%A A145369 _Wolfdieter Lang_, Oct 17 2008