Aleks Zigon Tankosic - cp's OEIS frontend

A373762 Row sums of A372208.

1, 217, 74817, 38539585, 28298839105, 28542001395321, 38373467110900417, 67090004454772224577, 149396282278530354664065, 416286791407819616695337305, 1429531867781567785731492164161, 5970020679482790599504907148094337, 29968790185674786127576736257632006337, 178970673260482220421285467858990357293945

Offset: 1

Aleks Zigon Tankosic, Jun 17 2024

nonn

A. Žigon Tankosič, The (l,r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A372208.

T := proc(n) local T, k; T := proc(n, k) option remember; if n = k then 1; elif k < 3 or n < k then 0; else T(n - 1, k - 1) + (n + k - 1)^3*T(n - 1, k); end if; end proc; add(T(n, k), k = 3 .. n); end proc;
seq(T(n), n = 0 .. 18);

A373761 Row sums of A371277.

Original entry on oeis.org

1, 65, 8281, 1832833, 648383681, 344889060481, 263241070687225, 277821953546314241, 393388714018670749633, 728818437848717556976321, 1729501321168395811504313561, 5161574686582090379099550582145, 19067391184605659825359397658612481, 85976151321068604971683536102183592193

Offset: 2

Aleks Zigon Tankosic, Jun 17 2024

nonn

A. Žigon Tankosič, The (l,r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A371277.

a := proc(n) local T, k; T := proc(n, k) option remember; if n = k then 1; elif k < 2 or n < k then 0; else T(n - 1, k - 1) + (n + k - 1)^3*T(n - 1, k); end if; end proc; add(T(n, k), k = 2 .. n); end proc:
seq(a(n), n = 0 .. 18);

A372208 Triangle read by rows, (3, 3)-Lah numbers.

Original entry on oeis.org

1, 216, 1, 74088, 728, 1, 37933056, 604800, 1728, 1, 27653197824, 642733056, 2904768, 3456, 1, 27653197824000, 883130895360, 5662172160, 10497600, 6200, 1, 36806406303744000, 1553703385006080, 13322923130880, 34467586560, 31422600, 10296, 1, 63601470092869632000, 3450292743162101760, 38111804456140800, 129651027770880, 163174556160, 82006848, 16128, 1

Offset: 3

Aleks Zigon Tankosic, Apr 22 2024

The (3, 3)-Lah numbers 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 the numbers 1, 2, 3 are in distinct lists, and 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).

The (3, 3)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=3 and r=3. More generally, the (m, r)-Lah numbers count ordered m-tuples (pi(1), pi(2), ..., pi(m)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, ..., r are in distinct lists, and bl(pi(1)) = bl(pi(2)) = ... = bl(pi(m)) where for i = {1, 2, ..., m} 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).

			Triangle begins:
              1;
            216,             1;
          74088,           728,           1;
       37933056,        604800,        1728,         1;
    27653197824,     642733056,     2904768,      3456,     1;
 27653197824000,  883130895360,  5662172160,  10497600,  6200,  1.
  ...
An example for T(4, 3). The corresponding partitions are
pi(1) = {(1),(2),(3,4)},
pi(2) = {(1),(2),(4,3)},
pi(3) = {(1),(3),(2,4)},
pi(4) = {(1),(3),(4,2)},
pi(5) = {(1,4),(2),(3)},
pi(6) = {(4,1),(2),(3)}, since A143498 for n=4, k=3 equals 6. Sets of their block leaders are bl(pi(1)) = bl(pi(2)) = bl(pi(3)) = bl(pi(4)) = bl(pi(5)) = bl(pi(6)) = {1,2,3}. Compute the number of ordered 3-tuples (i.e., ordered pairs) of partitions pi(1), pi(2), ..., pi(6) such that partitions in the same pair share the same set of block leaders. As there are six partitions with the set of block leaders equal to {1,2,3}, T(4, 3) = 6^3 = 216.

A. Žigon Tankosič, The (l,r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A143498, A371081, A371259, A371277.

T:= proc(n, k) option remember; `if`(k<3 or k>n, 0,
      `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^3*T(n-1, k)))
    end:
seq(seq(T(n, k), k=3..n), n=3..10);

A372208[n_, k_] := A372208[n, k] = Which[n < k || k < 3, 0, n == k, 1, True, A372208[n-1, k-1] + (n+k-1)^3*A372208[n-1, k]];
Table[A372208[n, k], {n, 3, 10}, {k, 3, n}] (* Paolo Xausa, Jun 11 2024 *)

A372208 = lambda n, k: 0 if (k < 3 or k > n) else (1 if (n == 3 and k == 3) else (A372208(n-1, k-1) + ((n + k - 1)**3) * A372208(n-1, k)))
print([A372208(n, k) for n in range(3, 11) for k in range(3, n+1)])

Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^3*T(n-1, k).
Explicit formula: T(n, k) = Sum_{4 <= j(1) < j(2) < ... < j(n-k) <= n} (2j(1)-2)^3 * (2j(2)-3)^3 * ... * (2j(n-k)-(n-k+1))^3.
Special cases:
T(n, k) = 0 for n < k or k < 3, [corrected by Paolo Xausa, Jun 11 2024]
T(n, n) = 1,
T(n, 3) = (A143498(n, 3))^3 = ((n+2)!)^3/1728000,
T(n, n-1) = 2^3 * Sum_{j=3..n-1} j^3.

A371489 Row sums of A371259.

Original entry on oeis.org

1, 37, 1865, 122961, 10278281, 1062714245, 133215991777, 19913088181377, 3500271565033393, 714944028588443461, 167950356430067342489, 44970653636552995400721, 13617086217015989835215993, 4630206565577103694905252965, 1756940670213816382344684605809, 739791104690854220148748269632001

Offset: 3

Aleks Zigon Tankosic, Apr 22 2024

nonn

Row sums of A371259 are the summed (2, 3)-Lah numbers (A371259).

A. Žigon Tankosič, The (l,r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A371259.

T := proc(n) local T, k; T := proc(n, k) option remember; if
n = k then 1; elif k < 3 or n < k then 0; else T(n - 1, k - 1) + (n + k -
1)^2*T(n - 1, k); end if; end proc; add(T(n, k), k = 3 .. n); end proc; seq(T(n), n = 0 .. 18);

A371488 Row sums of A371081.

Original entry on oeis.org

1, 17, 453, 17465, 921233, 63789145, 5616599013, 613148157073, 81298838448001, 12871080897739073, 2398329378160629861, 519554377953510437129, 129472180384695112970705, 36773246580917492621295817, 11807854666147122586977709125, 4255708041349122783137436409249, 1710617624877842754809697811363969

Offset: 2

Aleks Zigon Tankosic, Apr 22 2024

nonn

Row sums of A371081 are the summed (2, 2)-Lah numbers (A371081).

A. Žigon Tankosič, The (l,r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A371081.

T := proc(n) local T, k; T := proc(n, k) option remember; if n = k then 1; elif k < 2 or n < k then 0; else T(n - 1, k - 1) + (n + k - 1)^2*T(n - 1, k); end if; end proc; add(T(n, k), k = 2 .. n); end proc; seq(T(n), n = 0 .. 18);

A371277 Triangle read by rows, (3, 2)-Lah numbers.

Original entry on oeis.org

1, 64, 1, 8000, 280, 1, 1728000, 104040, 792, 1, 592704000, 54996480, 681408, 1792, 1, 303464448000, 40685137920, 736404480, 3066560, 3520, 1, 221225582592000, 40988602368000, 1020839500800, 6035420160, 10800000, 6264, 1, 221225582592000000, 54777055334400000, 1804999259750400, 14280657592320, 35670620160, 31941000, 10360, 1

Offset: 2

Aleks Zigon Tankosic, Mar 17 2024

The (3, 2)-Lah numbers 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 the numbers 1, 2 are in distinct lists, and 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).
The (3, 2)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=3 and r=2.
More generally, the (m, r)-Lah numbers count ordered m-tuples (pi(1), pi(2), ..., pi(m)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, ..., r are in distinct lists, and bl(pi(1)) = bl(pi(2)) = ... = bl(pi(m)) where for i = {1, 2, ..., m} 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).

			Triangle begins:
              1;
             64,              1;
           8000,            280,             1;
        1728000,         104040,           792,          1;
      592704000,       54996480,        681408,       1792,        1;
   303464448000,    40685137920,     736404480,    3066560,     3520,    1;
221225582592000, 40988602368000, 1020839500800, 6035420160, 10800000, 6264,  1.
 ...
An example for T(4, 3). The corresponding partitions are
pi(1) = {(1),(2),(3,4)},
pi(2) = {(1),(2),(4,3)},
pi(3) = {(1),(2,3),(4)},
pi(4) = {(1),(3,2),(4)},
pi(5) = {(1),(2,4),(3)},
pi(6) = {(1),(4,2),(3)},
pi(7) = {(1,3),(2),(4)},
pi(8) = {(3,1),(2),(4)},
pi(9) = {(1,4),(2),(3)},
pi(10) = {(4,1),(2),(3)}, since A143497 for n=4, k=3 equals 10. Sets of their block leaders are bl(pi(1)) = bl(pi(2)) = bl(pi(5)) = bl(pi(6)) = bl(pi(9)) = bl(pi(10)) = {1,2,3} and bl(pi(3)) = bl(pi(4)) = bl(pi(7)) = bl(pi(8)) = {1,2,4}.
Compute the number of ordered 3-tuples (i.e., ordered pairs) of partitions pi(1), pi(2), ..., pi(10) such that partitions in the same pair share the same set of block leaders. As there are six partitions with the set of block leaders equal to {1,2,3}, and four partitions with the set of block leaders equal to {1,2,4}, T(4, 3) = 6^3 + 4^3 = 280.

A. Žigon Tankosič, The (l, r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A143497, A371081, A371259, A372208.

T:= proc(n, k) option remember; `if`(k<2 or k>n, 0,
      `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^3*T(n-1, k)))
    end:
seq(seq(T(n, k), k=2..n), n=2..10);

A371277[n_, k_] := A371277[n, k] = Which[n < k || k < 2, 0, n == k, 1, True, A371277[n-1, k-1] + (n+k-1)^3*A371277[n-1, k]];
Table[A371277[n, k], {n, 2, 10}, {k, 2, n}] (* Paolo Xausa, Jun 12 2024 *)

A371277 = lambda n, k: 0 if (k < 2 or k > n) else (1 if (n == 2 and k == 2) else (A371277(n-1, k-1) + ((n + k - 1)**3) * A371277(n-1, k)))
print([A371277(n, k) for n in range(2, 10) for k in range(2, n+1)])

Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^3*T(n-1, k).
Explicit formula: T(n, k) = Sum_{3 <= j(1) < j(2) < ... < j(n-k) <= n} (2j(1)-2)^3 * (2j(2)-3)^3 * ... * (2j(n-k)-(n-k+1))^3.
Special cases:
T(n, k) = 0 for n < k or k < 2.
T(n, n) = 1.
T(n, 2) = (A143497(n,2))^3 = ((n+1)!)^3/216.
T(n, n-1) = 2^3 * Sum_{j=2..n-1} j^3.

A371259 Triangle read by rows, (2, 3)-Lah numbers.

Original entry on oeis.org

1, 36, 1, 1764, 100, 1, 112896, 9864, 200, 1, 9144576, 1099296, 34064, 344, 1, 914457600, 142159392, 6004512, 92200, 540, 1, 110649369600, 21385410048, 1156921920, 24075712, 213700, 796, 1, 15933509222400, 3724783667712, 248142106368, 6573957120, 78782912, 443744, 1120, 1

Offset: 3

Aleks Zigon Tankosic, Mar 16 2024

The (2, 3)-Lah numbers 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 the numbers 1, 2, 3 are in distinct lists, and 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 (2, 3)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=2 and r=3. More generally, the (m, r)-Lah numbers count ordered m-tuples (pi(1), pi(2), ..., pi(m)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, ..., r are in distinct lists, and bl(pi(1)) = bl(pi(2)) = ... = bl(pi(m)) where for i = {1, 2, ..., m} 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).

			Triangle begins:
           1;
          36,           1;
        1764,         100,          1;
      112896,        9864,        200,        1;
     9144576,     1099296,      34064,      344,     1;
   914457600,   142159392,    6004512,    92200,   540,  1;
110649369600, 21385410048, 1156921920, 24075712, 213700, 796, 1.
  ...
An example for T(4, 3). The corresponding partitions are
pi(1) = {(1),(2),(3,4)},
pi(2) = {(1),(2),(4,3)},
pi(3) = {(1),(3),(2,4)},
pi(4) = {(1),(3),(4,2)},
pi(5) = {(1,4),(2),(3)},
pi(6) = {(4,1),(2),(3)}, since A143498 for n=4, k=3 equals 6. Sets of their block leaders are bl(pi(1)) = bl(pi(2)) = bl(pi(3)) = bl(pi(4)) = bl(pi(5)) = bl(pi(6)) = {1,2,3}.
Compute the number of ordered 2-tuples (i.e., ordered pairs) of partitions pi(1), pi(2), ..., pi(6) such that partitions in the same pair share the same set of block leaders. As there are six partitions with the set of block leaders equal to {1,2,3}, T(4, 3) = 6^2 = 36.

A. Žigon Tankosič, The (l, r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Cf. A143498, A371081.

T:= proc(n, k) option remember; `if`(k<3 or k>n, 0,
      `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^2*T(n-1, k)))
    end:
seq(seq(T(n, k), k=3..n), n=3..10);

A371259[n_, k_] := A371259[n, k] = Which[n < k || k < 3, 0, n == k, 1, True, A371259[n-1, k-1] + (n+k-1)^2*A371259[n-1, k]];
Table[A371259[n, k], {n, 3, 10},{k, 3, n}] (* Paolo Xausa, Jun 11 2024 *)

A371259 = lambda n, k: 0 if (k < 3 or k > n) else (1 if (n == 3 and k == 3) else (A371259(n-1, k-1) + ((n + k - 1)**2) * A371259(n-1, k)))
print([A371259(n, k) for n in range(3, 11) for k in range(3, n+1)])

Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^2*T(n-1, k).
Explicit formula: T(n, k) = Sum_{4 <= j(1) < j(2) < ... < j(n-k) <= n} (2j(1)-2)^2 * (2j(2)-3)^2 * ... * (2j(n-k)-(n-k+1))^2.
Special cases:
T(n, k) = 0 for n < k or k < 3, [corrected by Paolo Xausa, Jun 11 2024]
T(n, n) = 1,
T(n, 3) = (A143498(n, 3))^2 = ((n+2)!)^2/14400,
T(n, n-1) = 2^2 * Sum_{j=3..n-1} j^2.

More terms from Michel Marcus, Jun 12 2025

A371081 Triangle read by rows, (2, 2)-Lah numbers.

Original entry on oeis.org

1, 16, 1, 400, 52, 1, 14400, 2948, 116, 1, 705600, 203072, 12344, 216, 1, 45158400, 17154432, 1437472, 38480, 360, 1, 3657830400, 1760601600, 191088544, 6978592, 99320, 556, 1, 365783040000, 216690624000, 29277351936, 1370470592, 26445312, 224420, 812, 1

Offset: 2

Aleks Zigon Tankosic, Mar 10 2024

The (2, 2)-Lah numbers 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 the numbers 1, 2 are in distinct lists, and 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 (2, 2)-Lah numbers T(n, k) are the (m, r)-Lah numbers for m=r=2. More generally, the (m, r)-Lah numbers count ordered m-tuples (pi(1), pi(2), ..., pi(m)) of partitions of the set {1, 2, ..., n} into k linearly ordered blocks (lists, for short) such that the numbers 1, 2, ..., r are in distinct lists, and bl(pi(1)) = bl(pi(2)) = ... = bl(pi(m)) where for i = {1, 2, ..., m} 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).

			Triangle begins:
         1;
        16,          1;
       400,         52,         1;
     14400,       2984,       116,       1;
    705600,     203072,     12344,     216,     1;
  45158400,    1715443,   1437472,   38480,   360,    1;
3657830400, 1760601600, 191088544,  6978592, 99320, 556, 1.
  ...
An example for T(4, 3). The corresponding partitions are
pi(1) = {(1),(2),(3,4)},
pi(2) = {(1),(2),(4,3)},
pi(3) = {(1),(2,3),(4)},
pi(4) = {(1),(3,2),(4)},
pi(5) = {(1),(2,4),(3)},
pi(6) = {(1),(4,2),(3)},
pi(7) = {(1,3),(2),(4)},
pi(8) = {(3,1),(2),(4)},
pi(9) = {(1,4),(2),(3)},
pi(10) = {(4,1),(2),(3)}, since A143497 for n=4, k=3 equals 10. Sets of their block leaders are bl(pi(1)) = bl(pi(2)) = bl(pi(5)) = bl(pi(6)) = bl(pi(9)) = bl(pi(10)) = {1,2,3} and
 bl(pi(3)) = bl(pi(4)) = bl(pi(7)) = bl(pi(8)) = {1,2,4}.
Compute the number of ordered 2-tuples (i.e., ordered pairs) of partitions pi(1), pi(2), ..., pi(10) such that partitions in the same pair share the same set of block leaders. As there are six partitions with the set of block leaders equal to {1,2,3}, and four partitions with the set of block leaders equal to {1,2,4}, T(4, 3) = 6^2 + 4^2 = 52.

A. Žigon Tankosič, The (l, r)-Lah Numbers, Journal of Integer Sequences, Article 23.2.6, vol. 26 (2023).

Column k=2 gives A001715(n+1)^2.

Cf. A143497, A371259, A371277, A371488 (row sums), A372208.

T:= proc(n, k) option remember; `if`(k<2 or k>n, 0,
      `if`(n=k, 1, T(n-1, k-1)+(n+k-1)^2*T(n-1, k)))
    end:
seq(seq(T(n, k), k=2..n), n=2..10);  # Alois P. Heinz, Mar 11 2024

A371081[n_, k_] := A371081[n, k] = Which[n < k || k < 2, 0, n == k, 1, True, A371081[n-1, k-1] + (n+k-1)^2*A371081[n-1, k]];
Table[A371081[n, k], {n, 2, 10}, {k, 2, n}] (* Paolo Xausa, Jun 12 2024 *)

A371081 = lambda n, k: 0 if (k < 2 or k > n) else (1 if (n == 2 and k == 2) else (A371081(n-1, k-1) + ((n + k - 1)**2) * A371081(n-1, k)))
print([A371081(n, k) for n in range(2, 10) for k in range(2, n+1)])

Recurrence relation: T(n, k) = T(n-1, k-1) + (n+k-1)^2*T(n-1, k).
Explicit formula: T(n, k) = Sum_{3 <= j(1) < j(2) < ... < j(n-k) <= n} (2j(1)-2)^2 * (2j(2)-3)^2 * ... * (2j(n-k)-(n-k+1))^2.
Special cases:
T(n, k) = 0 for n < k or k < 2,
T(n, n) = 1,
T(n, 2) = (A143497(n,2))^2 = A001715(n+1)^2 = ((n+1)!)^2/36,
T(n, n-1) = 2^2 * Sum_{j=2..n-1} j^2.

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User: Aleks Zigon Tankosic

A373762 Row sums of A372208.

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A372208 Triangle read by rows, (3, 3)-Lah numbers.

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A371489 Row sums of A371259.

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A371488 Row sums of A371081.

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A371277 Triangle read by rows, (3, 2)-Lah numbers.

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A371259 Triangle read by rows, (2, 3)-Lah numbers.

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A371081 Triangle read by rows, (2, 2)-Lah numbers.

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