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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 (list; graph; refs; listen; history; text; internal format)
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) =: M31(-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,...].

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,n-1)= [1,3,6,6,0,...], 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. 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

A049426 (row sums).

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

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Last modified April 21 20:41 EDT 2019. Contains 322328 sequences. (Running on oeis4.)