A145373 - OEIS

%I #10 Aug 29 2019 17:56:23

%S 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 285,45,5,1,0,6600,5420,1945,285,45,5,1,0,9600,23600,7920,1945,285,45,

%U 5,1,0,21600,66600,41100,11045,1945,285,45,5,1,0,14400,189600,151600,53600,11045

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

%C 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 The first column is [1,5,20,60,120,120,0,0,0,...]= A008279(5,n-1), n>=1.

%H W. Lang, <a href="/A145373/a145373.txt">First 10 rows of the array and more.</a>

%H W. 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 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 [1];[5,1];[20,5,1];[60,45,5,1];[120,160,45,5,1];...

%Y A145374 (row sums).

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_ Oct 17 2008