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A145370 Lower triangular array, called S1hat(-4), related to partition number array A145369.

%I A145370 #12 May 27 2025 10:31:53
%S A145370 1,4,1,12,4,1,24,28,4,1,24,72,28,4,1,0,264,136,28,4,1,0,384,456,136,
%T A145370 28,4,1,0,864,1344,712,136,28,4,1,0,576,4128,2112,712,136,28,4,1,0,
%U A145370 576,7488,7968,3136,712,136,28,4,1,0,0,13248,20544,11040,3136,712,136,28,4,1,0,0
%N A145370 Lower triangular array, called S1hat(-4), related to partition number array A145369.
%C A145370 If in the partition array M31hat(-4):=A145369 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(-4). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
%C A145370 The first column is [1,4,12,24,24,0,0,0,...]= A008279(4,n-1), n>=1.
%H A145370 Wolfdieter Lang, <a href="/A145370/a145370.txt">First 10 rows of the array and more.</a>
%H A145370 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 A145370 a(n,m) = sum(product(S1(-4;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, Y and e(n,m,q,j) is the exponent of j in the q-th m part partition of n. S1(-4,n,1)= A008279(4,n-1) = [1,4,12,24,24,0,0,0,...], n>=1.
%e A145370 Triangle begins:
%e A145370   [1];
%e A145370   [4,1];
%e A145370   [12,4,1];
%e A145370   [24,28,4,1];
%e A145370   [24,72,28,4,1];
%e A145370   ...
%Y A145370 Cf. A145371 (row sums).
%K A145370 nonn,easy,tabl
%O A145370 1,2
%A A145370 _Wolfdieter Lang_, Oct 17 2008