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A134273
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A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5).
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4
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1, 5, 1, 45, 15, 1, 585, 180, 75, 30, 1, 9945, 2925, 2250, 450, 375, 50, 1, 208845, 59670, 43875, 20250, 8775, 13500, 1875, 900, 1125, 75, 1, 5221125, 1461915, 1044225, 921375, 208845, 307125, 141750, 118125, 20475, 47250, 13125, 1575, 2625, 105, 1
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OFFSET
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1,2
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COMMENTS
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For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
Partition number array M_3(5), the k=5 member in the family of a generalization of the multinomial number arrays M_3 = M_3(1) = A036040.
The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
The S2(5,n,m):=A049029(n,m) numbers (generalized Stirling2 numbers) are obtained by summing in row n all numbers with the same part number m. In the same manner the S2(n,m) (Stirling2) numbers A008277 are obtained from the partition array M_3 = A036040.
a(n,k) enumerates unordered forests of increasing quintic (5-ary) trees related to the k-th partition of n in the A-St order. The m-forest is composed of m such trees, with m the number of parts of the partition.
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LINKS
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M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
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FORMULA
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a(n,k) = n!*Product_{j=1..n} (S2(5,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(5,n,1) = A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadruple- or 4-factorials) 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.
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EXAMPLE
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[1]; [51]; [45,15,1]; [585,180,75,30,1]; [9945,2925,2250,450,375,50,1]; ...
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CROSSREFS
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Cf. There are a(4, 3)=75=3*5^2 unordered 2-forest with 4 vertices, composed of two 5-ary increasing trees, each with two vertices: there are 3 increasing labelings (1, 2)(3, 4); (1, 3)(2, 4); (1, 4)(2, 3) and each tree comes in five versions from the 5-ary structure.
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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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