

A134278


A certain partition array in AbramowitzStegun order (ASt order), called M_3(6).


31



1, 6, 1, 66, 18, 1, 1056, 264, 108, 36, 1, 22176, 5280, 3960, 660, 540, 60, 1, 576576, 133056, 95040, 43560, 15840, 23760, 3240, 1320, 1620, 90, 1, 17873856, 4036032, 2794176, 2439360, 465696, 665280, 304920, 249480, 36960, 83160, 22680, 2310
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OFFSET

1,2


COMMENTS

For the ASt order of partitions see the AbramowitzStegun reference given in A117506.
Partition number array M_3(6), the k=6 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(6,n,m):=A049385(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 6ary trees related to the kth partition of n in the ASt order. The mforest is composed of m such trees, with m the number of parts of the partition.


LINKS

Table of n, a(n) for n=1..41.
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].
Wolfdieter Lang, First 10 rows and more.


FORMULA

a(n,k) = n!*Product_{j=1..n} (S2(6,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(6,n,1) = A049385(n,1) = A008548(n) = (5*n4)(!^5) (quintuple or 5factorials) and the exponent e(n,k,j) of j in the kth partition of n in the ASt ordering of the partitions of n. Exponents 0 can be omitted due to 0!=1.


EXAMPLE

[1]; [6,1]; [66,18,1]; [1056,264,108,36,1]; [22176,5280,3960,660,540,60,1]; ...
There are a(4,3) = 108 = 3*6^2 unordered 2forests with 4 vertices, composed of two 6ary 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 six versions from the 6ary structure.


CROSSREFS

Cf. A049412 (row sums, also of triangle A049385).
Cf. A134273 (M_3(5) partition array).
Sequence in context: A197655 A134279 A134280 * A049385 A266364 A009384
Adjacent sequences: A134275 A134276 A134277 * A134279 A134280 A134281


KEYWORD

nonn,easy,tabf


AUTHOR

Wolfdieter Lang, Nov 13 2007


STATUS

approved



