

A144877


Partition number array, called M31(3), related to A049410(n,m) = S1(3;n,m) (generalized Stirling triangle).


5



1, 3, 1, 6, 9, 1, 6, 24, 27, 18, 1, 0, 30, 180, 60, 135, 30, 1, 0, 0, 270, 360, 90, 1080, 405, 120, 405, 45, 1, 0, 0, 0, 1260, 0, 1890, 2520, 5670, 210, 3780, 2835, 210, 945, 63, 1, 0, 0, 0, 0, 1260, 0, 0, 10080, 11340, 30240, 0, 7560, 10080, 45360, 8505, 420, 10080, 11340
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OFFSET

1,2


COMMENTS

Each partition of n, ordered as in AbramowitzStegun (ASt order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31(3;n,k) with the kth partition of n in ASt order.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
First member (K=3) in the family M31(K) of partition number arrays.
If M31(3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(3) := A049410.


LINKS

Table of n, a(n) for n=1..62.
W. Lang, First 10 rows of the array and more.
W. Lang, Combinatorial Interpretation of Generalized Stirling Numbers, J. Int. Seqs. Vol. 12 (2009) 09.3.3.


FORMULA

a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(3;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(3;j,1)^e(n,k,j),j=1..n) with S1(3;n,1)= A008279(3,n1)= [1,3,6,6,0,...], n>=1 and the exponent e(n,k,j) of j in the kth partition of n in the ASt ordering of the partitions of n. M3(n,k)=A036040.


EXAMPLE

[1]; [3,1]; [6,9,1]; [6,24,27,18,1]; [0,30,180,60,135,30,1]; ...
a(4,3) = 27 = 3*S1(3;2,1)^2. The relevant partition of 4 is (2^2).


CROSSREFS

Cf. A049426 (row sums).
Cf. A144358 (M31(2) array), A144878 (M31(4) array).
Sequence in context: A306938 A257259 A074475 * A049410 A013610 A008573
Adjacent sequences: A144874 A144875 A144876 * A144878 A144879 A144880


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang Oct 09 2008, Oct 28 2008


STATUS

approved



