

A144279


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


4



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 AbramowitzStegun (ASt order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M32hat(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,...].
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



FORMULA

a(n,k)= product(S2(3,j,1)^e(n,k,j),j=1..n) with S2(3,n,1)= A008545(n1) = (4*n5)(!^4) (4factorials) for n>=2 and 1 if 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.
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



KEYWORD

nonn,easy,tabf


AUTHOR



STATUS

approved



