A134133 - OEIS

A134133

Triange T(n,k) read by rows: product of the factorials of the parts of the k-th partition of n listed in Abramowitz and Stegun order, 1 <= k <= A000041(n).

1, 1, 2, 1, 6, 2, 1, 24, 6, 4, 2, 1, 120, 24, 12, 6, 4, 2, 1, 720, 120, 48, 36, 24, 12, 8, 6, 4, 2, 1, 5040, 720, 240, 144, 120, 48, 36, 24, 24, 12, 8, 6, 4, 2, 1, 40320, 5040, 1440, 720, 576, 720, 240, 144, 96, 72, 120, 48, 36, 24, 16, 24, 12, 8, 6, 4, 2, 1, 362880, 40320, 10080

(list; graph; refs; listen; history; text; internal format)

OFFSET

0,3

COMMENTS

Partition number array M_3(2)= A130561 divided by partition number array M_3 = M_3(1) = A036040.

LINKS

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

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].

Wolfdieter Lang, First 10 rows and more.

FORMULA

T(n,k) = A130561(n,k)/A036040(n,k).

T(n,k) = product(j!^e(n,k,j),j=1..n) with 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.

T(n,k) = A321898(A185974(n,k)) = A112624(A036035(n,k)). - Andrew Howroyd, Oct 03 2025

EXAMPLE

Triangle begins:

0 | 1;

1 | 1;

2 | 2, 1;

3 | 6, 2, 1;

4 | 24, 6, 4, 2, 1;

5 | 120, 24, 12, 6, 4, 2, 1;

6 | 720, 120, 48, 36, 24, 12, 8, 6, 4, 2, 1;

...

PROG

(PARI) C(sig)={prod(k=1, #sig, sig[k]!)}

Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}

{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 03 2025

CROSSREFS

Cf. A069123 (version for reverse-lexicographically ordered partitions).

Cf. A134134 (triangle obtained by summing same m numbers).

Cf. A000041 (row lengths), A036035, A036036, A036040, A077365 (row sums), A112624, A130561, A185974, A321898.

Sequence in context: A335109 A179863 A069123 * A157392 A321352 A134134

Adjacent sequences: A134130 A134131 A134132 * A134134 A134135 A134136

KEYWORD

nonn,easy,tabf

AUTHOR

Wolfdieter Lang, Oct 12 2007

EXTENSIONS

Name improved and a(0) prepended by Andrew Howroyd, Oct 03 2025

STATUS

approved