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A134273 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5).

%I A134273 #19 Sep 25 2024 14:59:18
%S A134273 1,5,1,45,15,1,585,180,75,30,1,9945,2925,2250,450,375,50,1,208845,
%T A134273 59670,43875,20250,8775,13500,1875,900,1125,75,1,5221125,1461915,
%U A134273 1044225,921375,208845,307125,141750,118125,20475,47250,13125,1575,2625,105,1
%N A134273 A certain partition array in Abramowitz-Stegun order (A-St order), called M_3(5).
%C A134273 For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
%C A134273 Partition number array M_3(5), the k=5 member in the family of a generalization of the multinomial number arrays M_3 = M_3(1) = A036040.
%C A134273 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
%C A134273 The S2(5,n,m):=A049029(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 A134273 a(n,k) enumerates unordered forests of increasing quintic (5-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 A134273 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 A134273 Wolfdieter Lang, <a href="/A134273/a134273.txt">First 10 rows and more</a>.
%F A134273 a(n,k) = n!*Product_{j=1..n} (S2(5,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(5,n,1) = A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadruple- or 4-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 A134273 Triangle begins:
%e A134273   [1];
%e A134273   [51];
%e A134273   [45,15,1];
%e A134273   [585,180,75,30,1];
%e A134273   [9945,2925,2250,450,375,50,1];
%e A134273   ...
%Y A134273 Cf. There are a(4, 3)=75=3*5^2 unordered 2-forest with 4 vertices, composed of two 5-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 five versions from the 5-ary structure.
%Y A134273 Cf. A049120 (row sums also of triangle A049029).
%Y A134273 Cf. A134149 (M_3(4) array).
%K A134273 nonn,easy,tabf
%O A134273 1,2
%A A134273 _Wolfdieter Lang_, Nov 13 2007