A390434 - OEIS

%I #16 Nov 11 2025 22:19:55

%S 1,0,1,0,8,1,0,216,72,1,0,13824,9216,288,1,0,1728000,2004480,108000,

%T 800,1,0,373248000,689264640,57300480,691200,1800,1,0,128024064000,

%U 353276743680,42461314560,748500480,3087000,3528,1,0,65548320768000,257666770206720,42814591303680,1038715453440,6082836480,10838016,6272,1

%N Triangle read by rows, Lah numbers with level 3.

%C The Lah numbers with level 3 T(n, k) count ordered 3-tuples (pi(1), pi(2), pi(3)) of partitions of the set {1, ..., n} into k linearly ordered blocks (lists, for short) such that bl(pi(1)) = bl(pi(2) = bl(pi(3))) where for i = {1, 2, 3} 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).

%C The Lah numbers with level 3 T(n, k) are the Lah numbers with higher level for s=3. 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).

%H A. Žigon Tankosič, <a href="https://arxiv.org/abs/2510.26842">The Lah Numbers with Higher Level and the Lah Numbers of Order s</a>, arXiv:2510.26842 [math.GM], 2025.

%F Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^3*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.

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

%e Triangle begins:

%e   1

%e   0, 1

%e   0, 8, 1

%e   0, 216, 72, 1

%e   0, 13824, 9216, 288, 1

%e   0, 1728000, 2004480, 108000, 800, 1

%e   0, 373248000, 689264640, 57300480, 691200, 1800, 1

%e   0, 128024064000, 353276743680, 42461314560, 748500480, 3087000, 3528, 1

%e   ...

%p 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)^3*T(n - 1, k))); end proc;

%p for n from 0 to 10 do

%p     seq(T(n, k), k = 0 .. n);

%p end do;

%Y Cf. A371277, A269946, A390433.

%K nonn,tabl

%O 0,5

%A _Aleks Zigon Tankosic_, Nov 05 2025