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A144354
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Partition number array, called M31(4), related to A49352(n,m)= |S1(4;n,m)| (generalized Stirling triangle).
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4
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1, 4, 1, 20, 12, 1, 120, 80, 48, 24, 1, 840, 600, 800, 200, 240, 40, 1, 6720, 5040, 7200, 4000, 1800, 4800, 960, 400, 720, 60, 1, 60480, 47040, 70560, 84000, 17640, 50400, 28000, 33600, 4200, 16800, 6720, 700, 1680, 84, 1, 604800, 483840, 752640, 940800, 504000, 188160
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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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(4;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,...].
Fourth member (K=4) in the family M31(K) of partition number arrays.
If M31(4;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle |S1(4)|:= A049352.
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LINKS
| 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.
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FORMULA
| a(n,k)=(n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(|S1(4;j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S1(4;j,1)|^e(n,k,j),j=1..n) with |S1(4;n,1)|= A001715(n+2) = (n+2)!/3!, 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.
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EXAMPLE
| [1];[4,1];[20,12,1];[120,80,48,24,1];[840,600,800,200,240,40,1];...
a(4,3)= 48 = 3*|S1(4;2,1)|^2. The relevant partition of 4 is (2^2).
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CROSSREFS
| A049377 (row sums).
A144353 (M31(3) array), A144355 (M31(5) array).
Sequence in context: A143493 A062137 A143497 * A049352 A182867 A182826
Adjacent sequences: A144351 A144352 A144353 * A144355 A144356 A144357
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KEYWORD
| nonn,easy,tabf
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AUTHOR
| Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Oct 09 2008, Oct 28 2008
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