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A179233
Irregular triangle T(n,k) = A049019(n,k)/A096162(n,k), read by rows, 1<=k <= A000041(n).
2
1, 1, 1, 1, 1, 6, 1, 1, 8, 3, 18, 1, 1, 10, 20, 30, 45, 40, 1, 1, 12, 30, 10, 45, 360, 15, 80, 270, 75, 1, 1, 14, 42, 70, 63, 630, 210, 315, 140, 2520, 420, 175, 1050, 126, 1, 1, 16, 56, 112, 35, 84, 1008, 1680, 630, 840, 224, 5040, 1680, 10080, 105, 350, 11200, 4200, 336, 3150, 196, 1
OFFSET
0,6
COMMENTS
Rows have A000041(n) entries, with partitions in Abramowitz and Stegun order (A036036).
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..2713 (rows 0..20)
FORMULA
T(n,k) = A049019(n,k) / A096162(n,k) = A048996(n,k) * A036040(n,k).
Sum_{k=1..A000041(n)} T(n,k) = A120774(n).
EXAMPLE
Triangle begins:
0 | 1;
1 | 1;
2 | 1, 1;
3 | 1, 6, 1;
4 | 1, 8, 3, 18, 1;
5 | 1, 10, 20, 30, 45, 40, 1;
6 | 1, 12, 30, 10, 45, 360, 15, 80, 270, 75, 1;
7 | 1, 14, 42, 70, 63, 630, 210, 315, 140, 2520, 420, 175, 1050, 126, 1;
...
A049019(.,.) begins 1; 1; 2, 1; 6, 6, 1; 8, 6, 36, 24, ...
A096162(.,.) begins 1; 1; 2, 1; 1, 6, 1; 1, 2, 2, 24 ...
so
T(.,.) begins ..... 1; 1; 1, 1; 6, 1, 1; 8, 3, 18, 1 ...
PROG
(PARI)
C(sig)={my(S=Set(sig)); vecsum(sig)!*(#sig)!/prod(k=1, #sig, sig[k]!)/prod(k=1, #S, (#select(t->t==S[k], sig))!)^2}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 02 2025
CROSSREFS
Cf. A000041 (row lengths), A000670, A036036, A036040, A048996, A049019, A096162, A120774 (row sums).
Sequence in context: A171147 A171695 A382594 * A141600 A303489 A195408
KEYWORD
easy,nonn,tabf
AUTHOR
Alford Arnold, Jul 08 2010
EXTENSIONS
Extended, and bivariate indices restored - R. J. Mathar, Jul 13 2010
a(0)=1 prepended and a(58) onwards from Andrew Howroyd, Oct 02 2025
STATUS
approved