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A144879
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Partition number array, called M31(-5), related to A049411(n,m) = S1(-5;n,m) (generalized Stirling triangle).
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4
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1, 5, 1, 20, 15, 1, 60, 80, 75, 30, 1, 120, 300, 1000, 200, 375, 50, 1, 120, 720, 4500, 4000, 900, 6000, 1875, 400, 1125, 75, 1, 0, 840, 12600, 42000, 2520, 31500, 28000, 52500, 2100, 21000, 13125, 700, 2625, 105, 1, 0, 0, 16800, 134400, 126000, 3360, 100800, 336000
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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(-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, ...].
First member (K=5) in the family M31(-K) of partition number arrays.
If M31(-5;n,k) is summed over those k with fixed number of parts m one obtains the unsigned triangle S1(-5) := A049411.
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LINKS
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FORMULA
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a(n,k)=(n!/Product_{j=1..n} (e(n,k,j)!*j!^e(n,k,j))) * Product_{j=1..n} S1(-5;j,1)^e(n,k,j) = M3(n,k) * Product_{j=1..n} S1(-5;j,1)^e(n,k,j), with S1(-5;n,1) = A008279(5,n-1)= [1,5,20,60,120,120,0,...], 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]; [5,1]; [20,15,1]; [60,80,75,30,1]; [120,300,1000,200,375,50,1]; ...
a(4,3) = 75 = 3*S1(-5;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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