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A144270 Lower triangular array called S2hat(-1) related to partition number array A144269.

%I A144270 #17 Jul 01 2023 12:13:10
%S A144270 1,1,1,3,1,1,15,4,1,1,105,18,4,1,1,945,129,19,4,1,1,10395,1095,132,19,
%T A144270 4,1,1,135135,11880,1119,133,19,4,1,1,2027025,149940,12057,1122,133,
%U A144270 19,4,1,1,34459425,2218545,151560,12081,1123,133,19,4,1,1
%N A144270 Lower triangular array called S2hat(-1) related to partition number array A144269.
%C A144270 If in the partition array M32hat(-1)=A144269 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-1). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.
%C A144270 The first three columns are A001147, A144272, A144273.
%H A144270 Wolfdieter Lang, <a href="/A144270/a144270.txt">First 10 rows of the array and more</a>.
%H A144270 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 A144270 a(n,m)=sum(product(|S2(-1;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. |S2(-1;j,1)|= A001497(j-1,0) = A001147(j-1) = (2*j-3)(!^2) (2-factorials) for j>=2 and 1 if j=1.
%e A144270 Triangle begins
%e A144270     1;
%e A144270     1,  1;
%e A144270     3,  1, 1;
%e A144270    15,  4, 1, 1;
%e A144270   105, 18, 4, 1, 1;
%e A144270   ...
%Y A144270 Row sums A144271.
%K A144270 nonn,easy,tabl
%O A144270 1,4
%A A144270 _Wolfdieter Lang_, Oct 09 2008