1,2

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

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.

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

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.

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.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].

W. Lang, First 10 rows and more.

a(n,k)= n!*product((S2(5,j,1)/j!)^e(n,k,j)/e(n,k,j)!,j=1..n) with S2(5,n,1)=A049029(n,1) = A007696(n) = (4*n-3)(!^4) (quadrupel- 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.

[1]; [51]; [45,15,1]; [585,180,75,30,1]; [9945,2925,2250,450,375,50,1];...

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.

Cf. A049120 (row sums also of triangle A049029).

Cf. A134149 (M_3(4) array).

Sequence in context: A134274 A134275 A114154 this_sequence A048897 A049029 A051150

Adjacent sequences: A134270 A134271 A134272 this_sequence A134274 A134275 A134276

nonn,easy,tabf

Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de) Nov 13 2007