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A145367 Lower triangular array, called S1hat(-3), related to partition number array A145366.

%I A145367 #13 May 27 2025 10:30:34
%S A145367 1,3,1,6,3,1,6,15,3,1,0,24,15,3,1,0,54,51,15,3,1,0,36,108,51,15,3,1,0,
%T A145367 36,198,189,51,15,3,1,0,0,360,360,189,51,15,3,1,0,0,324,846,603,189,
%U A145367 51,15,3,1,0,0,216,1296,1332,603,189,51,15,3,1,0,0,216,2484,2754,2061,603,189
%N A145367 Lower triangular array, called S1hat(-3), related to partition number array A145366.
%C A145367 If in the partition array M31hat(-3):=A145366 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-3). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
%C A145367 The first column is [1,3,6,6,0,0,0,...]= A008279(3,n-1), n>=1.
%H A145367 Wolfdieter Lang, <a href="/A145367/a145367.txt">First 10 rows of the array and more.</a>
%H A145367 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 A145367 a(n,m) = sum(product(S1(-3;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(-3,n,1)= A008279(3,n-1) = [1,3,6,6,0,0,0,...], n>=1.
%e A145367 Triangle begins:
%e A145367   [1];
%e A145367   [3,1];
%e A145367   [6,3,1];
%e A145367   [6,15,3,1];
%e A145367   [0,24,15,3,1];
%e A145367   ...
%Y A145367 Cf. A145368 (row sums).
%K A145367 nonn,easy,tabl
%O A145367 1,2
%A A145367 _Wolfdieter Lang_, Oct 17 2008