A178803 Irregular triangle: T(n,k) is the factorial of the number of parts in the k-th partition of n listed in Abramowitz and Stegun order, read by rows, 1 <= k <= A000041(n).

1, 1, 1, 2, 1, 2, 6, 1, 2, 2, 6, 24, 1, 2, 2, 6, 6, 24, 120, 1, 2, 2, 2, 6, 6, 6, 24, 24, 120, 720, 1, 2, 2, 2, 6, 6, 6, 6, 24, 24, 24, 120, 120, 720, 5040, 1, 2, 2, 2, 2, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24, 120, 120, 120, 720, 720, 5040, 40320, 1, 2, 2, 2, 2, 6, 6, 6, 6, 6, 6, 6, 24, 24, 24, 24, 24

Offset

0,4

Comments

A036043(n,k) gives the number of parts in the k-th integer partition of n.

Links

Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)

Formula

T(n,k) = A036043(n,k)!.

T(n,k) = A130675(A185974(n,k)). - Andrew Howroyd, Oct 03 2025

Example

Triangle begins:
  0 | 1;
  1 | 1;
  2 | 1, 2;
  3 | 1, 2, 6;
  4 | 1, 2, 2, 6, 24;
  5 | 1, 2, 2, 6, 6, 24, 120;
  6 | 1, 2, 2, 2, 6, 6, 6, 24, 24, 120, 720;
  7 | 1, 2, 2, 2, 6, 6, 6, 6, 24, 24, 24, 120, 120, 720, 5040;
  ...
A036043 begins       0 1 1 2 1 2 3 1 2 2 3 4 1 2 2 3 3 4 5 ...
so this table begins 1 1 1 2 1 2 6 1 2 2 6 24 ...

Prog

(SageMath)
def A178803_row(n):
    return [factorial(len(p)) for k in (0..n) for p in Partitions(n, length=k)]
for n in (0..10): print(A178803_row(n)) # Peter Luschny, Nov 02 2019
(PARI)
Row(n)=[(#p)! | p<-partitions(n)]
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 03 2025

Crossrefs

Cf. A000041 (row lengths), A000142 (factorials), A036036, A036043, A101880 (row sums), A130675, A185974.

Sequence in context: A144358 A049404 A159885 * A292901 A083773 A129116

Adjacent sequences: A178800 A178801 A178802 * A178804 A178805 A178806

Keyword

easy,nonn,tabf

Author

Alford Arnold, Jun 17 2010

Extensions

a(0)=1 prepended and name edited by Andrew Howroyd, Oct 03 2025

Status

approved