A144280 - OEIS

%I #13 Jul 02 2023 12:48:07

%S 1,3,1,21,3,1,231,30,3,1,3465,294,30,3,1,65835,4599,321,30,3,1,

%T 1514205,81081,4788,321,30,3,1,40883535,1837836,84483,4869,321,30,3,1,

%U 1267389585,47609100,1892835,85050,4869,321,30,3,1,44358635475,1449052605,48681864

%N Lower triangular array called S2hat(-3) related to partition number array A144279.

%C If in the partition array M32khat(-3)= A144279 entries with the same parts number m are summed one obtains this triangle of numbers S2hat(-3). In the same way the Stirling2 triangle A008277 is obtained from the partition array M_3 = A036040.

%C The first three columns are A008545, A144282, A144283.

%H Wolfdieter Lang, <a href="/A144280/a144280.txt">First 10 rows of the array and more</a>.

%H 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 a(n,m) = Sum_{q=1..p(n,m)} Product_{j=1..n} |S2(-3;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(-3,n,1)|= A000369(n,1) = A008545(n-1) = (4*n-5)(!^4) (4-factorials) for n>=2 and 1 if n=1.

%e Triangle begins:

%e   [1];

%e   [3,1];

%e   [21,3,1];

%e   [231,30,3,1];

%e   [3465,294,30,3,1];

%e   ...

%Y Cf. A144279, A008277, A036040, A008284, A000369, A008545, A144282, A144283.

%Y Row sums A144281.

%Y Cf. A144275 (S2hat(-2)).

%K nonn,easy,tabl

%O 1,2

%A _Wolfdieter Lang_, Oct 09 2008