

A145356


Partition number array, called M31hat(6).


3



1, 6, 1, 42, 6, 1, 336, 42, 36, 6, 1, 3024, 336, 252, 42, 36, 6, 1, 30240, 3024, 2016, 1764, 336, 252, 216, 42, 36, 6, 1, 332640, 30240, 18144, 14112, 3024, 2016, 1764, 1512, 336, 252, 216, 42, 36, 6, 1, 3991680, 332640, 181440, 127008, 112896, 30240, 18144, 14112
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,2


COMMENTS

Each partition of n, ordered like in AbramowitzStegun (ASt order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(6;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,...].
Sixth member (K=6) in the family M31hat(K) of partition number arrays.
If M31hat(6;n,k) is summed over those k numerating partitions with fixed number of parts m one obtains the unsigned triangle S1hat(6):= A145357.


LINKS

Table of n, a(n) for n=1..52.
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) = product(S1(6;j,1)^e(n,k,j),j=1..n) with S1(6;n,1) = A049374(n,1) = A001725(n+4) = [1,6,42,336,3024,30240,332640,...] = (n+4)!/5!, 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.


EXAMPLE

[1];[6,1];[42,6,1];[336,42,36,6,1];[3024,336,252,42,36,6,1];...
a(4,3)= 36 = S1(6;2,1)^2. The relevant partition of 4 is (2^2).


CROSSREFS

A145358 (row sums).
A144890 (M31hat(5) array). A145357 (S1hat(6).
Sequence in context: A145927 A113365 A293172 * A145357 A035529 A135893
Adjacent sequences: A145353 A145354 A145355 * A145357 A145358 A145359


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang Oct 17 2008, Oct 28 2008


STATUS

approved



