A134283 - OEIS

%I #14 Aug 29 2019 17:45:35

%S 1,3,1,10,6,1,35,20,9,9,1,126,70,60,30,27,12,1,462,252,210,100,105,

%T 180,27,40,54,15,1,1716,924,756,700,378,630,300,270,140,360,108,50,90,

%U 18,1,6435,3432,2772,2520,1225,1386,2268,2100,945,900,504,1260,600,1080,81

%N A certain partition array in Abramowitz-Stegun (A-St)order, called M_0(3).

%C For the A-St order of partitions see the Abramowitz-Stegun reference given in A117506.

%C Partition number array M_0(3); the k=3 member in the family of a generalization of the multinomial number arrays M_0 = M_0(2) = A048996.

%C The sequence of row lengths is A000041 (partition numbers) [1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...].

%C The s2(3,n,m):=A035324(n,m) numbers (generalized Pascal triangle) are obtained by summing in row n all numbers with the same part number m. In the same manner the s2(2,n,m) = binomial(n-1,m-1) = A007318(n-1,m-1) numbers are obtained from the partition array M_0 = A048996.

%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="/A134283/a134283.txt">First 10 rows and more</a>.

%F a(n,k) = m!*Product_{j=1..n} (s2(3,j,1)^e(n,k,j))/e(n,k,j)! with s2(3,n,1) = A035324(n,1) = A001700(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 [1]; [3,1]; [10,6,1]; [35,20,9,9,1]; [126,70,60,30,27,12,1]; ...

%Y Cf. A049027 (row sums, also of triangle A035324).

%K nonn,easy,tabf

%O 1,2

%A _Wolfdieter Lang_, Nov 13 2007