A179313 Triangle T(n,k) read by rows: product of the compositorial weight of the k-th partition of n times A074664(.) applied to each part.

1, 1, 1, 2, 2, 1, 6, 4, 1, 3, 1, 22, 12, 4, 6, 3, 4, 1, 92, 44, 12, 4, 18, 12, 1, 8, 6, 5, 1, 426, 184, 44, 24, 66, 36, 12, 6, 24, 24, 4, 10, 10, 6, 1, 2146, 852, 184, 88, 36, 276, 132, 72, 18, 12, 88, 72, 24, 24, 1, 30, 40, 10, 12, 15, 7, 1, 11624, 4292, 852, 368, 264, 1278, 552

Offset

1,4

Comments

Row n has A000041(n) entries. T(n,k) is the product of A074664(a_i) over all parts a_i

multiplied by the compositorial weight A048996(n,k) of the k-th partition (Abramowitz-Stegun order)

of n = sum_i a_i.

Summing also over the partitions with a common number of parts would create A127743.

In row n=4, for example, the partitions 3+1 and 2+2, each with 2 parts, are represented by

T(4,2)=4 and T(4,3)= 1 here, and the sum 4+1=5 of the entries is the single entry A127743(4,.).

In this sense, the table is a refinement of A127743.

References

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, p. 831

Links

Table of n, a(n) for n=1..73.

Formula

T(n,k) = A048996(n,k) * A179380(n,k).

sum_{k=1..A000041(n)} T(n,k) = A000110(n).

Example

T(6,3) represents the 3rd partition of 6, namely 2+4. A074664(2)*A074664(4) = 1*6 is multiplied
by the weight A048996([2,4]) = 2!/1!/1! =2, and T(6,3) =1*6*2=12.
T(6,5) represents the 5th partition of 6, namely 1+1+4. A074664(1)*A074664(1)*A074664(4) = 1*1*6 is multiplied
by the weight A048996([1,1,4]) = 3!/2!/1! =3, and T(6,5) =1*1*6*3.
T(7,6) represents the 6th partition of 7, namely 1+2+4. A074664(1)*A074664(2)*A074664(4) = 1*1*6 is multiplied
the weight A048996([1,2,4]) = 3!/1!/1!/1! =6, and T(7,6) =1*1*6*6.
The triangle starts
1;
1,1;
2,2,1;
6,4,1,3,1;
22,12,4,6,3,4,1;
92,44,12,4,18,12,1,8,6,5,1;
426,184,44,24,66,36,12,6,24,24,4,10,10,6,1;

Crossrefs

Cf. A000041, A127743

Sequence in context: A290013 A220963 A375566 * A209692 A178886 A065052

Adjacent sequences: A179310 A179311 A179312 * A179314 A179315 A179316

Keyword

nonn,tabf

Author

Alford Arnold, Jul 11 2010

Extensions

Edited and extended by R. J. Mathar, Jul 16 2010

Status

approved