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A144275 Lower triangular array called S2hat(-2) related to partition number array A144274.

%I A144275 #14 Jul 02 2023 09:02:00
%S A144275 1,2,1,10,2,1,80,14,2,1,880,100,14,2,1,12320,1140,108,14,2,1,209440,
%T A144275 14880,1180,108,14,2,1,4188800,249280,15400,1196,108,14,2,1,96342400,
%U A144275 4801280,255400,15480,1196,108,14,2,1,2504902400,108574400,4888960,256440,15512
%N A144275 Lower triangular array called S2hat(-2) related to partition number array A144274.
%C A144275 If in the partition array M32khat(-2)= A144274 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-2). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.
%C A144275 The first three columns are A008544, A144277, A144278.
%H A144275 Wolfdieter Lang, <a href="/A144275/a144275.txt">First 10 rows of the array and more</a>.
%H A144275 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 A144275 a(n,m) = Sum_{q=1..p(n,m)} (Product_{j=1..n} |S2(-2;j,1)|^e(n,m,q,j)) 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. |S2(-2,n,1)|= A004747(n,1) = A008544(n-1) = (3*n-4)(!^3) (3-factorials) for n>=2 and 1 if n=1.
%e A144275 Triangle begins:
%e A144275   [1];
%e A144275   [2,1];
%e A144275   [10,2,1];
%e A144275   [80,14,2,1];
%e A144275   [880,100,14,2,1];
%e A144275   ...
%Y A144275 Row sums A144276.
%Y A144275 A144270 (S2hat(-1)).
%Y A144275 Cf. A004747, A008284, A008544.
%K A144275 nonn,easy,tabl
%O A144275 1,2
%A A144275 _Wolfdieter Lang_, Oct 09 2008