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A134278 A certain partition array in Abramowitz-Stegun order (A-St 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

For the A-St order of partitions see the Abramowitz-Stegun 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 6-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.

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].

W. Lang, First 10 rows and more.

FORMULA

a(n,k)= n!*product((S2(6,j,1)/j!)^e(n,k,j)/e(n,k,j)!,j=1..n) with S2(6,n,1)=A049385(n,1) = A008548(n) = (5*n-4)(!^5) (quintupel- or 5-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.

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 2-forest with 4 vertices, composed of two 6-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 six versions from the 6-ary structure.

CROSSREFS

Cf. A049412 (row sums, also of triangle A049385).

Cf. A134273 (M_3(5) partition array).

Sequence in context: A197655 A134279 A134280 * A049385 A009384 A051151

Adjacent sequences:  A134275 A134276 A134277 * A134279 A134280 A134281

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang Nov 13 2007

STATUS

approved

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Last modified December 22 06:09 EST 2014. Contains 252328 sequences.