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A145364 Lower triangular array, called S1hat(-2), related to partition number array A145363.

%I A145364 #13 May 27 2025 10:30:42
%S A145364 1,2,1,2,2,1,0,6,2,1,0,4,6,2,1,0,4,12,6,2,1,0,0,12,12,6,2,1,0,0,8,28,
%T A145364 12,6,2,1,0,0,8,24,28,12,6,2,1,0,0,0,24,56,28,12,6,2,1,0,0,0,16,56,56,
%U A145364 28,12,6,2,1,0,0,0,16,48,120,56,28,12,6,2,1,0,0,0,0,48,112,120,56,28,12,6,2,1
%N A145364 Lower triangular array, called S1hat(-2), related to partition number array A145363.
%C A145364 If in the partition array M31hat(-2):=A145363 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-2). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
%C A145364 The first column is [1,2,2,0,0,0,...]= A008279(2,n-1), n>=1.
%H A145364 Wolfdieter Lang, <a href="/A145364/a145364.txt">First 10 rows of the array and more.</a>
%H A145364 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 A145364 a(n,m) = sum(product(S1(-2;j,1)^e(n,m,q,j),j=1..n),q=1..p(n,m)) if n>=m>=1, else 0. Here p(n,m)=A008284(n,m), the number of m parts partitions of n and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-2,n,1)= A008279(2,n-1) = [1,2,2,0,0,0,...], n>=1.
%e A145364 Triangle begins:
%e A145364   [1];
%e A145364   [2,1];
%e A145364   [2,2,1];
%e A145364   [0,6,2,1];
%e A145364   [0,4,6,2,1];
%e A145364   ...
%Y A145364 Cf. A145365 (row sums).
%K A145364 nonn,easy,tabl
%O A145364 1,2
%A A145364 _Wolfdieter Lang_, Oct 17 2008