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A144357
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Partition number array, called M31(-1), related to A049403(n,m)= S1(-1;n,m) (generalized Stirling triangle).
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4
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1, 1, 1, 0, 3, 1, 0, 0, 3, 6, 1, 0, 0, 0, 0, 15, 10, 1, 0, 0, 0, 0, 0, 0, 15, 0, 45, 15, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 105, 0, 105, 21, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 105, 0, 0, 420, 0, 210, 28, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 945, 0, 0, 1260, 0, 378, 36
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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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(-1;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=1) in the family M31(-K) of partition number arrays.
If M31(-1;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-1):= A049403.
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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(-1;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-1;j,1)^e(n,k,j),j=1..n) with S1(-1;n,1)|= A008279(1,n-1)= [1,1,0,...], 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];[1,1];[0,3,1];[0,0,3,6,1];[0,0,0,0,15,10,1];...
a(4,3)= 3 = 3*S1(-1;2,1)^2. The relevant partition of 4 is (2^2).
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CROSSREFS
| A000085 (row sums).
A144358 (M31(-2) array).
Sequence in context: A035654 A170846 A085604 * A122848 A054548 A059202
Adjacent sequences: A144354 A144355 A144356 * A144358 A144359 A144360
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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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