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A134149
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A certain partition array in Abramowitz-Stegun (A-St) order.
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4
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1, 4, 1, 28, 12, 1, 280, 112, 48, 24, 1, 3640, 1400, 1120, 280, 240, 40, 1, 58240, 21840, 16800, 7840, 4200, 6720, 960, 560, 720, 60, 1, 1106560, 407680, 305760, 274400, 76440, 117600, 54880, 47040, 9800, 23520, 6720, 980, 1680, 84, 1, 24344320
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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(4), the k=4 member of a family of generalizations of the multinomial number array 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(4,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 quaternary 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(4,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(4,n,1) = A035469(n,1) = A007559(n) = (3*n-2)!!! (triple- or 3-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]; [4,1]; [28,12,1]; [280,112,48,24,1]; [3640,1400,1120,280,240,40,1]; ...
a(4,3)=48 from the third (k=3) partition (2^2) of 4: 4!*((4/2!)^2)/2 = 48, because S2(4,2,1) = 4!!! = 4*1 = 4.
There are a(4,3) = 48 = 3*4^2 unordered 2-forests with 4 vertices, composed of two increasing quaternary (4-ary) 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 four versions from the quaternary structure.
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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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