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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 Abramowitz-Stegun (A-St order; for the reference see A134278), is mapped to a nonnegative integer a(n,k) =: M31hat(6;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,...].

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 k-th partition of n in the A-St 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

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Last modified January 26 11:22 EST 2021. Contains 340438 sequences. (Running on oeis4.)