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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
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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(-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
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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(-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
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[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
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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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