VOOZH about

URL: https://oeis.org/A144357

⇱ A144357 - OEIS

login
A144357
Partition number array, called M31(-1), related to A049403(n,m) = S1(-1;n,m) (generalized Stirling triangle).
4
1, 1, 1, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 0, 15, 10, 1, 0, 0, 0, 0, 0, 0, 15, 0, 45, 15, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 105, 0, 105, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 105, 0, 0, 420, 0, 210, 28, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 945, 0, 0, 1260, 0, 378, 36
OFFSET
1,5
COMMENTS
Each partition of n, ordered as in Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(-1;n,k) with the k-th partition of n in A-St order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
First member (K=1) in the family M31(-K) of partition number arrays.
If M31(-1;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-1) := A049403.
LINKS
FORMULA
a(n,k) = (n!/(Product_{j=1..n} e(n,k,j)!*j!^e(n,k,j))*Product_{j=1..n} S1(-1;j,1)^e(n,k,j) = M3(n,k)*Product_{j=1..n} S1(-1;j,1)^e(n,k,j) with S1(-1;n,1) |= A008279(1,n-1) = [1,1,0,...], n >= 1 and the exponent e(n,k,j) of j in the k-th partition of n in the A-St ordering of the partitions of n. M3(n,k) = A036040.
EXAMPLE
[1]; [1,1]; [0,3,1]; [0,0,3,6,1]; [0,0,0,0,15,10,1]; ...
a(4,3) = 3 = 3*S1(-1;2,1)^2. The relevant partition of 4 is (2^2).
CROSSREFS
Cf. A000085 (row sums).
Cf. A144358 (M31(-2) array).
Sequence in context: A306268 A354490 A365970 * A122848 A386635 A272481
KEYWORD
nonn,easy,tabf
AUTHOR
Wolfdieter Lang, Oct 09 2008, Oct 28 2008
STATUS
approved