A144279 - OEIS

A144279

Partition number array, called M32hat(-3)= 'M32(-3)/M3'= 'A143173/A036040', related to A000369(n,m)= |S2(-3;n,m)| (generalized Stirling triangle).

1, 3, 1, 21, 3, 1, 231, 21, 9, 3, 1, 3465, 231, 63, 21, 9, 3, 1, 65835, 3465, 693, 441, 231, 63, 27, 21, 9, 3, 1, 1514205, 65835, 10395, 4851, 3465, 693, 441, 189, 231, 63, 27, 21, 9, 3, 1, 40883535, 1514205, 197505, 72765, 53361, 65835, 10395, 4851, 2079, 1323, 3465

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OFFSET

1,2

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) =: M32hat(-3;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,...].

If M32hat(-3;n,k) is summed over those k with fixed number of parts m one obtains triangle S2hat(-3):= A144280(n,m).

LINKS

Table of n, a(n) for n=1..55.

Wolfdieter Lang, First 10 rows of the array and more.

Wolfdieter Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.

FORMULA

a(n,k) = Product_{j=1..n} |S2(-3,j,1)|^e(n,k,j), with |S2(-3,n,1)|= A008545(n-1) = (4*n-5)(!^4) (4-factorials) for n>=2 and 1 if 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.

Formally a(n,k)= 'M32(-3)/M3' = 'A143173/A036040' (elementwise division of arrays).

EXAMPLE

a(4,3) = 9 = |S2(-3,2,1)|^2. The relevant partition of 4 is (2^2).

CROSSREFS

Cf. A036040, A143173, A134278, A000041, A008545.

Cf. A144274 (M32hat(-2) array), A144284 (M32hat(-4) array).

Sequence in context: A334910 A016531 A221365 * A144280 A107717 A143173

Adjacent sequences: A144276 A144277 A144278 * A144280 A144281 A144282

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Oct 09 2008

STATUS

approved