login
A181512
Irregular triangle T(n,k) = A181415(n,k) / A036040(n,k), read by rows, 1 <= k <= A000041(n).
3
1, 1, 1, 1, 2, 2, 1, 3, 3, 6, 6, 1, 4, 4, 12, 12, 24, 24, 1, 5, 5, 5, 20, 20, 20, 60, 60, 120, 120, 1, 6, 6, 6, 30, 30, 30, 30, 120, 120, 120, 360, 360, 720, 720, 1, 7, 7, 7, 7, 42, 42, 42, 42, 42, 210, 210, 210, 210, 210, 840, 840, 840, 2520, 2520, 5040, 5040
OFFSET
1,5
COMMENTS
Rows have A000041(n) entries, with partitions in Abramowitz and Stegun order (A036036).
This is an irregular table related to labeled rooted trees and to Bell numbers.
A181511 contains the same values, without repetition.
LINKS
Andrew Howroyd, Table of n, a(n) for n = 1..2713 (rows 1..20)
FORMULA
Sum_{k=1.. A000041(n)} T(n,k) = A181513(n).
T(n,k) = A178888(n,k) / n. - Andrew Howroyd, Oct 02 2025
EXAMPLE
Triangle begins:
1;
1,1;
1,2,2;
1,3,3,6,6;
1,4,4,12,12,24,24;
1,5,5,5,20,20,20,60,60,120,120;
1,6,6,6,30,30,30,30,120,120,120,360,360,720,720;
PROG
(PARI)
C(sig)={(binomial(vecsum(sig), #sig)) * (#sig)! / vecsum(sig)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=1, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 02 2025
CROSSREFS
Cf. A000041 (row lengths), A036036, A036040, A181415, A181511, A181513 (row sums).
Sequence in context: A374434 A372688 A011020 * A076019 A071453 A212306
KEYWORD
nonn,tabf
AUTHOR
Alford Arnold, Oct 26 2010
EXTENSIONS
a(57) onwards from Andrew Howroyd, Oct 02 2025
STATUS
approved