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A145373 Lower triangular array, called S1hat(-5), related to partition number array A145372.

%I A145373 #13 May 27 2025 10:35:17
%S A145373 1,5,1,20,5,1,60,45,5,1,120,160,45,5,1,120,820,285,45,5,1,0,1920,1320,
%T A145373 285,45,5,1,0,6600,5420,1945,285,45,5,1,0,9600,23600,7920,1945,285,45,
%U A145373 5,1,0,21600,66600,41100,11045,1945,285,45,5,1,0,14400,189600,151600,53600,11045
%N A145373 Lower triangular array, called S1hat(-5), related to partition number array A145372.
%C A145373 If in the partition array M31hat(-5):=A145372 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-5). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
%C A145373 The first column is [1,5,20,60,120,120,0,0,0,...]= A008279(5,n-1), n>=1.
%H A145373 Wolfdieter Lang, <a href="/A145373/a145373.txt">First 10 rows of the array and more.</a>
%H A145373 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 A145373 a(n,m) = sum(product(S1(-5;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(-5,n,1)= A008279(5,n-1) = [1,5,20,60,120,120,0,0,0,...], n>=1.
%e A145373 Triangle begins:
%e A145373   [1];
%e A145373   [5,1];
%e A145373   [20,5,1];
%e A145373   [60,45,5,1];
%e A145373   [120,160,45,5,1];
%e A145373   ...
%Y A145373 Cf. A145374 (row sums).
%K A145373 nonn,easy,tabl
%O A145373 1,2
%A A145373 _Wolfdieter Lang_, Oct 17 2008