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%I #14 Aug 29 2019 16:35:05
%S 1,4,1,28,12,1,280,112,48,24,1,3640,1400,1120,280,240,40,1,58240,
%T 21840,16800,7840,4200,6720,960,560,720,60,1,1106560,407680,305760,
%U 274400,76440,117600,54880,47040,9800,23520,6720,980,1680,84,1,24344320
%N A certain partition array in Abramowitz-Stegun (A-St) order.
%C For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
%C Partition number array M_3(4), the k=4 member of a family of generalizations of the multinomial number array 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(4,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 quaternary 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 W. Lang, <a href="/A134149/a134149.txt">First 10 rows and more. </a>
%F a(n,k) = n!*Product_{j=1..n} (S2(4,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(4,n,1) = A035469(n,1) = A007559(n) = (3*n-2)!!! (triple- or 3-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]; [4,1]; [28,12,1]; [280,112,48,24,1]; [3640,1400,1120,280,240,40,1]; ...
%e a(4,3)=48 from the third (k=3) partition (2^2) of 4: 4!*((4/2!)^2)/2 = 48, because S2(4,2,1) = 4!!! = 4*1 = 4.
%e There are a(4,3) = 48 = 3*4^2 unordered 2-forests with 4 vertices, composed of two increasing quaternary (4-ary) 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 four versions from the quaternary structure.
%Y Cf. A134144 (M_3(3) array).
%K nonn,easy,tabf
%O 1,2
%A _Wolfdieter Lang_, Nov 13 2007
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