A390433 Triangle read by rows, Lah numbers with level 2.

1, 0, 1, 0, 4, 1, 0, 36, 20, 1, 0, 576, 536, 56, 1, 0, 14400, 19872, 3280, 120, 1, 0, 518400, 988128, 229792, 13000, 220, 1, 0, 25401600, 63758592, 19601280, 1529792, 39620, 364, 1, 0, 1625702400, 5189847552, 2023886592, 204706112, 7235072, 101136, 560, 1

Offset

0,5

Comments

The Lah numbers with level 2 T(n, k) count ordered 2-tuples (pi(1), pi(2)) of partitions of the set {1, ..., n} into k linearly ordered blocks (lists, for short) such that bl(pi(1)) = bl(pi(2)) where for i = {1, 2} and pi(i) = b(1)^i, b(2)^i, ..., b(k)^i, where b(1)^i, b(2)^i, ..., b(k)^i are the blocks of partition pi(i), bl(pi(i)) = {min(b(1))^i, min(b(2))^i, ..., min(b(k))^i} is the set of block leaders, i.e., of minima of the lists in partition pi(i).

The Lah numbers with level 2 T(n, k) are the Lah numbers with higher level for s=2. More generally, the Lah numbers with higher level count ordered s-tuples (pi(1), pi(2), ..., pi(s)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that bl(pi(1)) = bl(pi(2)) = ... = bl(pi(s)) where for i = {1, 2, ..., s} and pi(i) = {b(1)^i, b(2)^i, ..., b(k)^i}, where b(1)^i, b(2)^i,..., b(k)^i are the blocks of partition pi(i), bl(pi(i)) = {min(b(1))^i, min(b(2))^i, ..., min (b(k))^i} is the set of block leaders, i.e., of minima of the lists in partition pi(i).

Links

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

A. Žigon Tankosič, The Lah Numbers with Higher Level and the Lah Numbers of Order s, arXiv:2510.26842 [math.GM], 2025.

Formula

Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^2*T(n-1, k) with boundary conditions T(n,n) = 1, T(n,0) = T(0,n) = 0 for n > 0, T(n, k) = 0 for n < k.

Explicit formula: T(n, k) = Sum_{1 <= j(1) < j(2) < ... < j(n-k) <= n} (2*j(1)-2)^2 * (2*j(2)-3)^2 * ... * (2*j(n-k)-(n-k+1))^2.

Example

Triangle begins:
  1
  0, 1
  0, 4, 1
  0, 36, 20, 1
  0, 576, 536, 56, 1
  0, 14400, 19872, 3280, 120, 1
  0, 518400, 988128, 229792, 13000, 220, 1
  0, 25401600, 63758592, 19601280, 1529792, 39620, 364, 1
  0, 1625702400, 5189847552, 2023886592, 204706112, 7235072, 101136, 560, 1
  0, 131681894400, 520610457600, 250080125184, 31501566720, 1427433280, 27057728, 227136, 816, 1.
  ...

Maple

T := proc(n, k) option remember; `if`(n = k, 1, `if`(k < 0 or n < k, 0, T(n - 1, k - 1) + (n +k- 1)^2*T(n - 1, k))); end proc;
for n from 0 to 10 do
    seq(T(n, k), k = 0 .. n); end do;

Crossrefs

Cf. A371081, A268434, A390434.

Sequence in context: A395929 A371080 A130636 * A299354 A117414 A085639

Adjacent sequences: A390430 A390431 A390432 * A390434 A390435 A390436

Keyword

nonn,tabl

Author

Aleks Zigon Tankosic, Nov 05 2025

Status

approved