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A144358
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Partition number array, called M31(-2), related to A049404(n,m)= S1(-2;n,m) (generalized Stirling triangle).
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5
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1, 2, 1, 2, 6, 1, 0, 8, 12, 12, 1, 0, 0, 40, 20, 60, 20, 1, 0, 0, 0, 40, 0, 240, 120, 40, 180, 30, 1, 0, 0, 0, 0, 0, 0, 280, 840, 0, 840, 840, 70, 420, 42, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2240, 0, 0, 1120, 6720, 1680, 0, 2240, 3360, 112, 840, 56, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2240, 0, 0
(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(-2;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=2) in the family M31(-K) of partition number arrays.
If M31(-2;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-2):= A049404.
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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(-2;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-2;j,1)^e(n,k,j),j=1..n) with S1(-2;n,1)|= A008279(2,n-1)= [1,2,2,0,...], 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];[2,1];[2,6,1];[0,8,12,12,1];[0,0,40,20,60,20,1];...
a(4,3)= 12 = 3*S1(-2;2,1)^2. The relevant partition of 4 is (2^2).
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CROSSREFS
| A049425 (row sums).
A144357 (M31(-1) array), A144877 (M31(-3) array).
Sequence in context: A000619 A006602 A144824 * A049404 A159885 A178803
Adjacent sequences: A144355 A144356 A144357 * A144359 A144360 A144361
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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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