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%I #12 Sep 01 2018 21:32:09
%S 1,6,1,66,18,1,1056,264,108,36,1,22176,5280,3960,660,540,60,1,576576,
%T 133056,95040,43560,15840,23760,3240,1320,1620,90,1,17873856,4036032,
%U 2794176,2439360,465696,665280,304920,249480,36960,83160,22680,2310
%N A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(6).
%C For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
%C 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.
%C The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
%C 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.
%C 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.
%H M. Abramowitz and I. A. Stegun, eds., <a href="http://www.convertit.com/Go/ConvertIt/Reference/AMS55.ASP">Handbook of Mathematical Functions</a>, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
%H Wolfdieter Lang, <a href="/A134278/a134278.pdf">First 10 rows and more.</a>
%F 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*n-4)(!^5) (quintuple- 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.
%e [1]; [6,1]; [66,18,1]; [1056,264,108,36,1]; [22176,5280,3960,660,540,60,1]; ...
%e There are a(4,3) = 108 = 3*6^2 unordered 2-forests 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.
%Y Cf. A049412 (row sums, also of triangle A049385).
%Y Cf. A134273 (M_3(5) partition array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Nov 13 2007
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