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A144877
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Partition number array, called M31(-3), related to A049410(n,m) = S1(-3;n,m) (generalized Stirling triangle).
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5
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1, 3, 1, 6, 9, 1, 6, 24, 27, 18, 1, 0, 30, 180, 60, 135, 30, 1, 0, 0, 270, 360, 90, 1080, 405, 120, 405, 45, 1, 0, 0, 0, 1260, 0, 1890, 2520, 5670, 210, 3780, 2835, 210, 945, 63, 1, 0, 0, 0, 0, 1260, 0, 0, 10080, 11340, 30240, 0, 7560, 10080, 45360, 8505, 420, 10080, 11340
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OFFSET
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1,2
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COMMENTS
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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(-3;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=3) in the family M31(-K) of partition number arrays.
If M31(-3;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-3) := A049410.
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LINKS
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FORMULA
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a(n,k)= (n!/product(e(n,k,j)!*j!^(e(n,k,j),j=1..n))*product(S1(-3;j,1)^e(n,k,j),j=1..n) = M3(n,k)*product(S1(-3;j,1)^e(n,k,j),j=1..n) with S1(-3;n,1)|= A008279(3,n-1)= [1,3,6,6,0,...], 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
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[1]; [3,1]; [6,9,1]; [6,24,27,18,1]; [0,30,180,60,135,30,1]; ...
a(4,3) = 27 = 3*S1(-3;2,1)^2. The relevant partition of 4 is (2^2).
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CROSSREFS
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KEYWORD
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nonn,easy,tabf
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AUTHOR
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STATUS
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approved
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