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A145361 Characteristic partition array for partitions with parts 1 and 2 only.

%I A145361 #13 May 27 2025 10:30:49
%S A145361 1,1,1,0,1,1,0,0,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,0,1,0,1,1,1,0,0,0,0,0,
%T A145361 0,0,0,0,0,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,
%U A145361 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,1,0,1,1,1,0,0,0,0,0,0,0,0,0
%N A145361 Characteristic partition array for partitions with parts 1 and 2 only.
%C A145361 Each partition of n, ordered like in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to 1 if the partition has parts 1 or 2 only and to 0 otherwise.
%C A145361 First member (K=1) in the family M31hat(-K) of partition number arrays.
%C A145361 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42,...].
%C A145361 This array is array A144357 divided entrywise by the array M_3=M3(1)=A036040. Formally 'A144357/A036040'. E.g. a(4,3)= 1 = 3/3 = A144357(4,3)/A036040(4,3).
%C A145361 If M31hat(-1;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(-1):= A145362 .
%H A145361 Wolfdieter Lang, <a href="/A145361/a145361.txt">First 10 rows of the array and more.</a>
%H A145361 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 A145361 a(n,k) = product(S1(-1;j,1)^e(n,k,j),j=1..n) with S1(-1;n,1) = A008279(1,n-1) = [1,1,0,0,0,...], 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 A145361 Triangle begins:
%e A145361   [1];
%e A145361   [1,1];
%e A145361   [0,1,1];
%e A145361   [0,0,1,1,1];
%e A145361   [0,0,0,0,1,1,1];
%e A145361   ...
%e A145361 a(4,3)= 1 = S1(-1;2,1)^2. The relevant partition of 4 is (2^2).
%Y A145361 Cf. A145363 (M31hat(-2)).
%K A145361 nonn,easy,tabf
%O A145361 1,1
%A A145361 _Wolfdieter Lang_, Oct 17 2008