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A144268
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Partition number array, called M32(-5), related to A013988(n,m)= |S2(-5;n,m)| ( generalized Stirling triangle).
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4
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1, 5, 1, 55, 15, 1, 935, 220, 75, 30, 1, 21505, 4675, 2750, 550, 375, 50, 1, 623645, 129030, 70125, 30250, 14025, 16500, 1875, 1100, 1125, 75, 1, 21827575, 4365515, 2258025, 1799875, 451605, 490875, 211750, 144375, 32725, 57750, 13125, 1925, 2625, 105, 1, 894930575
(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)=:M32(-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,...].
a(n,k) enumerates special unordered forests related to the k-th partition of n in the A-St order. The k-th partition of n is given by the exponents enk =(e(n,k,1),...,e(n,k,n)) of 1,2,...n. The number of parts is m = sum(e(n,k,j),j=1..n). The special (enk)-forest is composed of m rooted increasing (r+4)-ary trees if the outdegree is r>=0.
If M32(-5;n,k) is summed over those k with fixed number of parts m one obtains triangle A013988(n,m)= |S2(-5;n,m)|, a generalization of Stirling numbers of the second kind. For S2(K;n,m), K from the integers, see the reference under A035342.
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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(|S2(-5,j,1)|^e(n,k,j),j=1..n) = M3(n,k)*product(|S2(-5,j,1)|^e(n,k,j),j=1..n), with |S2(-5,n,1)|= A008543(n-1) = (6*n-7)(!^6) (6-factorials) for n>=2 and 1 if 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. Exponents 0 can be omitted due to 0!=1. M3(n,k):= A036040(n,k), k=1..p(n), p(n):= A000041(n).
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EXAMPLE
| a(4,3)=75. The relevant partition of 4 is (2^2). The 75 unordered (0,2,0,0)-forests are composed of the following 2 rooted increasing trees 1--2,3--4; 1--3,2--4 and 1--4,2--3. The trees are 5-ary because r=1 vertices are 5-ary and for the leaves (r=0) the arity does not matter. Each of the three differently labelled forests comes therefore in 5^2=25 versions due to the two 5-ary root vertices.
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CROSSREFS
| A144267 (M32(-4) array).
Sequence in context: A051150 A144341 A144342 * A013988 A050970 A138548
Adjacent sequences: A144265 A144266 A144267 * A144269 A144270 A144271
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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
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