This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A134144 #14 Aug 28 2019 17:58:49
%S A134144 1,3,1,15,9,1,105,60,27,18,1,945,525,450,150,135,30,1,10395,5670,4725,
%T A134144 2250,1575,2700,405,300,405,45,1,135135,72765,59535,55125,19845,33075,
%U A134144 15750,14175,3675,9450,2835,525,945,63,1,2027025,1081080,873180,793800
%N A134144 A certain partition array in Abramowitz-Stegun order (A-St order).
%C A134144 For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.
%C A134144 Partition number array M_3(3), the k=3 member of a family of generalizations of the multinomial number array M_3 = M_3(1) = A036040.
%C A134144 The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].
%C A134144 The S2(3,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 A134144 a(n,k) enumerates unordered forests of increasing ternary trees related to the k-th partition of n in the A-St order. The forest is composed of m such trees, with m the number of parts of the partition.
%H A134144 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 A134144 W. Lang, <a href="/A134144/a134144.txt">First 10 rows and more.</a>
%F A134144 a(n,k) = n!*Product_{j=1..n} (S2(3,j,1)/j!)^e(n,k,j)/e(n,k,j)! with S2(3,n,1) = A035342(n,1) = A001147(n) = (2*n-1)!! 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 A134144 [1]; [3,1]; [15,9,1]; [105,60,27,18,1]; [945,525,450,150,135,30,1]; ...
%e A134144 a(4,3)=27 from the partition (2^2) of 4: 4!*((3/2!)^2)/2! = 27.
%e A134144 There are a(4,3) = 27 = 3*3^2 unordered 2-forests with 4 vertices, composed of two increasing ternary 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 three versions from the ternary structure.
%Y A134144 Cf. A049118 (row sums, identical with those of triangle A035342).
%K A134144 nonn,easy,tabf
%O A134144 1,2
%A A134144 _Wolfdieter Lang_, Nov 13 2007