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A145357 Lower triangular array, called S1hat(6), related to partition number array A145356.

%I A145357 #13 May 27 2025 10:30:53
%S A145357 1,6,1,42,6,1,336,78,6,1,3024,588,78,6,1,30240,6804,804,78,6,1,332640,
%T A145357 62496,8316,804,78,6,1,3991680,753984,85176,9612,804,78,6,1,51891840,
%U A145357 8273664,1021608,94248,9612,804,78,6,1,726485760,109118016,11394432,1157688,102024
%N A145357 Lower triangular array, called S1hat(6), related to partition number array A145356.
%C A145357 If in the partition array M31hat(6):=A145356 entries belonging to partitions with the same parts number m are summed one obtains this triangle of numbers S1hat(6). In the same way the signless Stirling1 triangle |A008275| is obtained from the partition array M_2 = A036039.
%C A145357 The first columns are A001725(n+4), A145359, A145360,...
%H A145357 Wolfdieter Lang, <a href="/A145357/a145357.txt">First 10 rows of the array and more.</a>
%H A145357 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 A145357 a(n,m) = sum(product(|S1(6;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(6,n,1)|= A049374(n,1) = A001725(n+4) = (n+4)!/5!.
%e A145357 Triangle begins:
%e A145357   [1];
%e A145357   [6,1];
%e A145357   [42,6,1];
%e A145357   [336,78,6,1];
%e A145357   [3024,588,78,6,1];
%e A145357   ...
%Y A145357 Cf. A145358 (row sums).
%K A145357 nonn,easy,tabl
%O A145357 1,2
%A A145357 _Wolfdieter Lang_, Oct 17 2008