A119443 Triangle T(n,k) read by rows: number of ordered lists of unordered partitions whose sums are given by the k-th partition of n listed in Abramowitz and Stegun order, 1 <= k <= A000041(n).

1, 1, 2, 1, 3, 4, 1, 5, 6, 4, 6, 1, 7, 10, 12, 9, 12, 8, 1, 11, 14, 20, 9, 15, 36, 8, 12, 24, 10, 1, 15, 22, 28, 30, 21, 60, 27, 36, 20, 72, 32, 15, 40, 12, 1, 22, 30, 44, 42, 25, 33, 84, 90, 60, 54, 28, 120, 54, 144, 16, 25, 120, 80, 18, 60, 14, 1

Offset

0,3

Comments

This triangle is a refinement of the triangle A060642.

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].

Formula

T(n,k) = A119441(n,k) * A048996(n,k).

Example

Triangle begins:
  0 | 1;
  1 | 1;
  2 | 2, 1;
  3 | 3, 4, 1;
  4 | 5, 6, 4, 6, 1;
  5 | 7, 10, 12, 9, 12, 8, 1;
  6 | 11, 14, 20, 9, 15, 36, 8, 12, 24, 10, 1;
  7 | 15, 22, 28, 30, 21, 60, 27, 36, 20, 72, 32, 15, 40, 12, 1;
  ...
T(6,4) = 9 corresponds to the partition {3,3}. Each of the two parts can independently be one of 1+1+1, 1+2 or 3+3 giving a total of 3*3 = 9.

Prog

(PARI)
C(sig)={my(S=Set(sig)); prod(k=1, #sig, numbpart(sig[k]))*(#sig)!/prod(k=1, #S, (#select(t->t==S[k], sig))!)}
Row(n)={apply(C, [Vecrev(p) | p<-partitions(n)])}
{ for(n=0, 7, print(Row(n))) } \\ Andrew Howroyd, Oct 03 2025

Crossrefs

Cf. A000041 (row lengths), A036036, A048996, A055887 (row sums), A060642, A119441, A179314.

Sequence in context: A395917 A390496 A078753 * A209413 A126198 A055888

Adjacent sequences: A119440 A119441 A119442 * A119444 A119445 A119446

Keyword

nonn,easy,look,tabf

Author

Alford Arnold, May 22 2006

Extensions

More terms from R. J. Mathar, Jul 12 2013

Name improved and a(0)=1 prepended by Andrew Howroyd, Oct 03 2025

Status

approved