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A144355 Partition number array, called M31(5), related to A049353(n,m)= |S1(5;n,m)| (generalized Stirling triangle). 4
1, 5, 1, 30, 15, 1, 210, 120, 75, 30, 1, 1680, 1050, 1500, 300, 375, 50, 1, 15120, 10080, 15750, 9000, 3150, 9000, 1875, 600, 1125, 75, 1, 151200, 105840, 176400, 220500, 35280, 110250, 63000, 78750, 7350, 31500, 13125, 1050, 2625, 105, 1, 1663200, 1209600, 2116800 (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(5;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,...].

Fifth member (K=5) in the family M31(K) of partition number arrays.

If M31(5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(5)|:= A049353.

LINKS

Table of n, a(n) for n=1..47.

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(5;j,1)|^e(n,k,j),j=1..n)= M3(n,k)*product(|S1(5;j,1)|^e(n,k,j),j=1..n) with |S1(5;n,1)|= A001720(n+3) = (n+3)!/4!, 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];[5,1];[30,15,1];[210,120,75,30,1];[1680,1050,1500,300,375,50,1];...

a(4,3)= 75 = 3*|S1(5;2,1)|^2. The relevant partition of 4 is (2^2).

CROSSREFS

A049378 (row sums).

A144354 (M31(4) array), A144356 (M31(6) array).

Sequence in context: A049460 A145926 A062140 * A049353 A165226 A027759

Adjacent sequences:  A144352 A144353 A144354 * A144356 A144357 A144358

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang Oct 09 2008, Oct 28 2008

STATUS

approved

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Last modified October 17 10:50 EDT 2017. Contains 293469 sequences.