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A134144 A certain partition array in Abramowitz-Stegun order (A-St order). +0
4
1, 3, 1, 15, 9, 1, 105, 60, 27, 18, 1, 945, 525, 450, 150, 135, 30, 1, 10395, 5670, 4725, 2250, 1575, 2700, 405, 300, 405, 45, 1, 135135, 72765, 59535, 55125, 19845, 33075, 15750, 14175, 3675, 9450, 2835, 525, 945, 63, 1, 2027025, 1081080, 873180, 793800 (list; graph; listen)
OFFSET

1,2

COMMENT

For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.

Partition number array M_3(3), the k=3 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(3,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 ternary trees related to the k-th partition of n in the A-St order. The forest is composed of m such trees, with m the number of parts of the partition.

LINKS

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, December 1972 [alternative scanned copy].

W. Lang, First 10 rows and more.

FORMULA

a(n,k)= n!*product((S2(3,j,1)/j!)^e(n,k,j)/e(n,k,j)!,j=1..n) with S2(3,n,1)=A035342(n,1) = A001147(n) = (2*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. Exponents 0 can be omitted due to 0!=1.

EXAMPLE

[1]; [3,1]; [15,9,1]; [105,60,27,18,1]; [945,525,450,150,135,30,1];...

a(4,3)=27 from the partition (2^2) of 4: 4!*((3/2!)^2)/2! = 27.

There are a(4,3)=27= 3*3^2 unordered 2-forest with 4 vertices, composed of two increasing ternary trees, each with two vertices: there are 3 increasing labellings (1,2)(3,4); (1,3)(2,4); (1,4)(2,3) and each tree comes in three versions from the ternary structure.

CROSSREFS

Cf. A049118 (row sums, identical with those of triangle A035342).

Sequence in context: A113389 A038553 A135896 this_sequence A035342 A039815 A134685

Adjacent sequences: A134141 A134142 A134143 this_sequence A134145 A134146 A134147

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Nov 13 2007

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Last modified July 23 10:48 EDT 2008. Contains 142285 sequences.


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