A176565 Triangle read by rows: T(n,k) = binomial(P(n)+k,k) + binomial(P(n)+n-k,n-k) - binomial(P(n)+n,n) where P(n) = A000041(n) is the number of partitions of n.

1, 1, 1, 1, 0, 1, 1, -6, -6, 1, 1, -64, -84, -64, 1, 1, -454, -636, -636, -454, 1, 1, -7996, -10933, -11648, -10933, -7996, 1, 1, -116264, -154904, -165852, -165852, -154904, -116264, 1, 1, -4292122, -5475909, -5769895, -5823025, -5769895, -5475909, -4292122, 1

Offset

0,8

Comments

Triangle is symmetric.

Links

Table of n, a(n) for n=0..44.

Example

Triangle begins:
  {1},
  {1, 1},
  {1, 0, 1},
  {1, -6, -6, 1},
  {1, -64, -84, -64, 1},
  { 1, -454, -636, -636, -454, 1},
  {1, -7996, -10933, -11648, -10933, -7996, 1},
  ...

Maple

T:= (n, k)-> ((C, P)->  C(P(n)+k, k)+C(P(n)+n-k, n-k)-C(P(n)+n, n))(binomial, combinat[numbpart]):
seq(seq(T(n, k), k=0..n), n=0..8);  # Alois P. Heinz, Mar 23 2026

Mathematica

t[n_, m_] = Binomial[PartitionsP[n] + m, m] + Binomial[PartitionsP[n] + n - m, n - m] - (Binomial[PartitionsP[n] + 0, 0] + Binomial[PartitionsP[ n] + n - 0, n - 0]) + 1;
Table[Table[t[n, m], {m, 0, n}], {n, 0, 10}];
Flatten[%]

Crossrefs

Cf. A000041.

Sequence in context: A205457 A155868 A322622 * A176567 A372272 A283100

Adjacent sequences: A176562 A176563 A176564 * A176566 A176567 A176568

Keyword

sign,tabl,less

Author

Roger L. Bagula, Apr 20 2010

Extensions

Edited by Sean A. Irvine, Mar 23 2026

Status

approved