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A144878
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Partition number array, called M31(-4), related to A049424(n,m)= S1(-4;n,m) (generalized Stirling triangle).
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5
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1, 4, 1, 12, 12, 1, 24, 48, 48, 24, 1, 24, 120, 480, 120, 240, 40, 1, 0, 144, 1440, 1440, 360, 2880, 960, 240, 720, 60, 1, 0, 0, 2016, 10080, 504, 10080, 10080, 20160, 840, 10080, 6720, 420, 1680, 84, 1, 0, 0, 0, 16128, 20160, 0, 16128, 80640, 80640, 161280, 1344, 40320
(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,...].
First 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):= A049424.
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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)= A008279(4,n-1)= [1,4,12,12,24,24,0,...], 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.
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EXAMPLE
| [1];[4,1];[12,12,1];[24,48,48,24,1];[24,120,480,120,240,40,1];...
a(4,3)= 48 = 3*S1(-4;2,1)^2. The relevant partition of 4 is (2^2).
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CROSSREFS
| A049427 (row sums).
A144877 (M31(-3) array), A144879 (M31(-5) array).
Sequence in context: A106194 A051290 A125105 * A049424 A157394 A078219
Adjacent sequences: A144875 A144876 A144877 * A144879 A144880 A144881
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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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