A117401 -id:A117401 - cp's OEIS frontend

A176627 Triangle T(n, k) = 12^(k*(n-k)), read by rows.

1, 1, 1, 1, 12, 1, 1, 144, 144, 1, 1, 1728, 20736, 1728, 1, 1, 20736, 2985984, 2985984, 20736, 1, 1, 248832, 429981696, 5159780352, 429981696, 248832, 1, 1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins as:
  1;
  1,       1;
  1,      12,           1;
  1,     144,         144,             1;
  1,    1728,       20736,          1728,             1;
  1,   20736,     2985984,       2985984,         20736,           1;
  1,  248832,   429981696,    5159780352,     429981696,      248832,       1;
  1, 2985984, 61917364224, 8916100448256, 8916100448256, 61917364224, 2985984, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000326,
Cf. A118190 (q=2), this sequence (q=3), A176631 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), this sequence (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

[(12)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021

(* First program *)
T[n_, k_, q_]= (Binomial[3*q,2]/3)^(k*(n-k));
Table[T[n,k,3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 30 2021 *)
(* Second program *)
With[{m=10}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

flatten([[(12)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, k) = Product_{j=1..n} (q*(3*q - 1)/2)^j and q = 3.
T(n, k, q) = (binomial(3*q, 2)/3)^(k*(n-k)) with q = 3.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 10. - G. C. Greubel, Jun 30 2021

Edited by G. C. Greubel, Jun 30 2021

A176639 Triangle T(n, k) = 15^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 15, 1, 1, 225, 225, 1, 1, 3375, 50625, 3375, 1, 1, 50625, 11390625, 11390625, 50625, 1, 1, 759375, 2562890625, 38443359375, 2562890625, 759375, 1, 1, 11390625, 576650390625, 129746337890625, 129746337890625, 576650390625, 11390625, 1

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins as:
  1;
  1,      1;
  1,     15,          1;
  1,    225,        225,           1;
  1,   3375,      50625,        3375,          1;
  1,  50625,   11390625,    11390625,      50625,      1;
  1, 759375, 2562890625, 38443359375, 2562890625, 759375, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000384.
Cf. A158116 (q=2), this sequence (q=3), A176641 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), this sequence (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

[(15)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021

(* First program *)
T[n_, k_, q_] = Binomial[2*q, 2]^(k*(n-k));
Table[T[n, k, 3], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 30 2021 *)
(* Second program *)
With[{m=13}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

flatten([[(15)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, k) = Product_{j=1..n} (q*(2*q - 1))^j and q = 3.
T(n, k, q) = binomial(2*q, 2)^(k*(n-k)) with q = 3.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 13. - G. C. Greubel, Jun 30 2021

Edited by G. C. Greubel, Jun 30 2021

A176642 Triangle T(n, k) = 8^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 8, 1, 1, 64, 64, 1, 1, 512, 4096, 512, 1, 1, 4096, 262144, 262144, 4096, 1, 1, 32768, 16777216, 134217728, 16777216, 32768, 1, 1, 262144, 1073741824, 68719476736, 68719476736, 1073741824, 262144, 1, 1, 2097152, 68719476736, 35184372088832, 281474976710656, 35184372088832, 68719476736, 2097152, 1

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins as:
  1;
  1,      1;
  1,      8,          1;
  1,     64,         64,           1;
  1,    512,       4096,         512,           1;
  1,   4096,     262144,      262144,        4096,          1;
  1,  32768,   16777216,   134217728,    16777216,      32768,      1;
  1, 262144, 1073741824, 68719476736, 68719476736, 1073741824, 262144, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. this sequence (q=2), A176643 (q=3), A176644 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), this sequence (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).
Cf. A000567, A004247.

[8^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021

T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 2], {n, 0, 12}, {k, 0, n}]//Flatten
With[{m=6}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

flatten([[8^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 2.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 2.
T(n, k) = 8^A004247(n,k), where A004247 is interpreted as a triangle. [relation detected by sequencedb.net]. - R. J. Mathar, Jun 30 2021
T(n, k, m) = (m+2)^(k*(n-k)) with m = 6. - G. C. Greubel, Jun 30 2021

Edited by R. J. Mathar and G. C. Greubel, Jun 30 2021

A176631 Triangle T(n, k) = 22^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 22, 1, 1, 484, 484, 1, 1, 10648, 234256, 10648, 1, 1, 234256, 113379904, 113379904, 234256, 1, 1, 5153632, 54875873536, 1207269217792, 54875873536, 5153632, 1, 1, 113379904, 26559922791424, 12855002631049216, 12855002631049216, 26559922791424, 113379904, 1

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins as:
  1;
  1,       1;
  1,      22,           1;
  1,     484,         484,             1;
  1,   10648,      234256,         10648,           1;
  1,  234256,   113379904,     113379904,      234256,       1;
  1, 5153632, 54875873536, 1207269217792, 54875873536, 5153632, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000326.
Cf. A118190 (q=2), A176627 (q=3), this sequence (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), this sequence (m=20), A176641 (m=26), A176644 (m=38).

[22^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 01 2021

T[n_, k_, q_]= (Binomial[3*q,2]/3)^(k*(n-k)); Table[T[n,k,4], {n,0,12}, {k,0,n}]//Flatten
Table[22^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 01 2021 *)

flatten([[22^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 01 2021

T(n, k, q) = c(n, q)/(c(k, q)*c(n-k, q)), where c(n, k) = Product_{j=1..n} (q*(3*q - 1)/2)^j and q = 4.
T(n, k, q) = (binomial(3*q, 2)/3)^(k*(n-k)) with q = 4.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 20. - G. C. Greubel, Jul 01 2021

Edited by G. C. Greubel, Jul 01 2021

A176641 Triangle T(n, k) = 28^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 28, 1, 1, 784, 784, 1, 1, 21952, 614656, 21952, 1, 1, 614656, 481890304, 481890304, 614656, 1, 1, 17210368, 377801998336, 10578455953408, 377801998336, 17210368, 1, 1, 481890304, 296196766695424, 232218265089212416, 232218265089212416, 296196766695424, 481890304, 1

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins as:
  1;
  1,        1;
  1,       28,            1;
  1,      784,          784,              1;
  1,    21952,       614656,          21952,            1;
  1,   614656,    481890304,      481890304,       614656,        1;
  1, 17210368, 377801998336, 10578455953408, 377801998336, 17210368, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000384.
Cf. A158116 (q=2), A176639 (q=3), this sequence (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), this sequence (m=26).
Cf. A007318 (p=0), A118180 (p=1), A158116 (p=2), A158117 (p=3), A176639 (p=4), A176643 (p=5), this sequence (p=6).

[(28)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 30 2021

T[n_, k_, q_] = Binomial[2*q, 2]^(k*(n-k));
Table[T[n, k, 4], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 30 2021 *)
With[{m=26}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

flatten([[(28)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, k) = Product_{j=1..n} (q*(2*q - 1))^j and q = 4.
From G. C. Greubel, Jun 30 2021: (Start)
T(n, k, q) = binomial(2*q, 2)^(k*(n-k)) with q = 4.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 26.
T(n, k, p) = binomial(p+2, 2)^(k*(n-k)) with p = 6. (End)

Edited by G. C. Greubel, Jun 30 2021

A176643 Triangle T(n, k) = 21^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 21, 1, 1, 441, 441, 1, 1, 9261, 194481, 9261, 1, 1, 194481, 85766121, 85766121, 194481, 1, 1, 4084101, 37822859361, 794280046581, 37822859361, 4084101, 1, 1, 85766121, 16679880978201, 7355827511386641, 7355827511386641, 16679880978201, 85766121, 1

Offset: 0

Roger L. Bagula, Apr 22 2010

			Triangle begins as:
  1;
  1,       1;
  1,      21,           1;
  1,     441,         441,            1;
  1,    9261,      194481,         9261,           1;
  1,  194481,    85766121,     85766121,      194481,       1;
  1, 4084101, 37822859361, 794280046581, 37822859361, 4084101, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Cf. A000567.
Cf. A176642 (q=2), this sequence (q=3), A176644 (q=4).
Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), this sequence (m=19), A176631 (m=20), A176641 (m=26).

[(21)^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 01 2021

T[n_, k_, q_]:= (q*(3*q-2))^(k*(n-k)); Table[T[n, k, 3], {n, 0, 12}, {k, 0, n}]//Flatten
Table[21^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jul 01 2021 *)

flatten([[(21)^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jul 01 2021

T(n, k, q) = c(n,q)/(c(k, q)*c(n-k, q)) where c(n, q) = (q*(3*q - 2))^binomial(n+1,2) and q = 3.
T(n, k, q) = (q*(3*q-2))^(k*(n-k)) with q = 3.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 19. - G. C. Greubel, Jul 01 2021

Edited by G. C. Greubel, Jul 01 2021

A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.

Original entry on oeis.org

1, 1, 2, 1, 4, 16, 1, 8, 64, 512, 1, 16, 256, 4096, 65536, 1, 32, 1024, 32768, 1048576, 33554432, 1, 64, 4096, 262144, 16777216, 1073741824, 68719476736, 1, 128, 16384, 2097152, 268435456, 34359738368, 4398046511104, 562949953421312

Offset: 0

Mohammad K. Azarian, May 10 2021

T(n, k) is the number of relations from an n-element set into a k-element set, n >= 0, 0 <= k <= n.

T(n,k) is the size of the right principal ideal generated by A where A is an n X n matrix over GF(2) having rank k. The right principal ideal of A contains precisely the matrices whose image is contained in the image of A. - Geoffrey Critzer, Sep 25 2022

			T(3,3) = number of relations from a 3-element set into a 3-element set=2^(3*3)=512.
Triangle begins:
   1
   1   2
   1   4      16
   1   8      64      512
   1  16     256     4096      65536
   1  32    1024    32768    1048576    33554432
   ...

Michael De Vlieger, Table of n, a(n) for n = 0..1325 (rows n = 0..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.

Cf. A000079, A089072, A008292, A031971, A117401, A344260 (row sums).

Cf. A022166, A288853.

Mathematica
```
Table[2^(n*k), {n, 0, 10}, {k, 0, n}]
```

T(n,k) = 2^(n*k).

T(n,k) = Sum_{j=0..k} A288853(n,j)*A022166(n,j). - Geoffrey Critzer, Jan 02 2023

A368220 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal and vertical reflections by an asymmetric tile.

Original entry on oeis.org

1, 6, 6, 16, 76, 16, 72, 1056, 1056, 72, 256, 16576, 65536, 16576, 256, 1056, 262656, 4196352, 4196352, 262656, 1056, 4096, 4197376, 268435456, 1073790976, 268435456, 4197376, 4096, 16512, 67117056, 17180000256, 274878431232, 274878431232, 17180000256, 67117056, 16512

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k |    1       2           3              4                  5
  ----+-----------------------------------------------------------
    1 |    1       6          16             72                256
    2 |    6      76        1056          16576             262656
    3 |   16    1056       65536        4196352          268435456
    4 |   72   16576     4196352     1073790976       274878431232
    5 |  256  262656   268435456   274878431232    281474976710656
    6 | 1056 4197376 17180000256 70368756760576 288230376688582656

Peter Kagey, Illustration of T(2,2)=76
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023. See also J. Int. Seq., (2024) Vol. 27, Art. No. 24.6.1, pp. A-1, A-3.

Cf. A117401, A225910, A368218, A368219, A368222, A368224.

A368220[n_, m_] := 2^(n*m - 2)*(2^(n*m) + Boole[EvenQ[n*m]] + Boole[EvenQ[n]] + Boole[EvenQ[m]])

A368222 Table read by downward antidiagonals: T(n,k) is the number of tilings of the n X k grid up to horizontal reflection by an asymmetric tile.

Original entry on oeis.org

1, 2, 3, 4, 10, 4, 8, 36, 32, 10, 16, 136, 256, 136, 16, 32, 528, 2048, 2080, 512, 36, 64, 2080, 16384, 32896, 16384, 2080, 64, 128, 8256, 131072, 524800, 524288, 131328, 8192, 136, 256, 32896, 1048576, 8390656, 16777216, 8390656, 1048576, 32896, 256

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  1    2      4       8        16          32
   2 |  3   10     36     136       528        2080
   3 |  4   32    256    2048     16384      131072
   4 | 10  136   2080   32896    524800     8390656
   5 | 16  512  16384  524288  16777216   536870912
   6 | 36 2080 131328 8390656 536887296 34359869440

Peter Kagey, Illustration of T(3,2)=32
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A368220, A368221, A368224, A117401.

A368222[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n]])

A368224 Table read by antidiagonals: T(n,k) is the number of tilings of the n X k grid up to 180-degree rotation by an asymmetric tile.

Original entry on oeis.org

1, 3, 3, 4, 10, 4, 10, 36, 36, 10, 16, 136, 256, 136, 16, 36, 528, 2080, 2080, 528, 36, 64, 2080, 16384, 32896, 16384, 2080, 64, 136, 8256, 131328, 524800, 524800, 131328, 8256, 136, 256, 32896, 1048576, 8390656, 16777216, 8390656, 1048576, 32896, 256

Offset: 1

Peter Kagey, Dec 18 2023

			Table begins:
  n\k|  1    2      3       4         5           6
  ---+---------------------------------------------
   1 |  1    3      4      10        16          36
   2 |  3   10     36     136       528        2080
   3 |  4   36    256    2080     16384      131328
   4 | 10  136   2080   32896    524800     8390656
   5 | 16  528  16384  524800  16777216   536887296
   6 | 36 2080 131328 8390656 536887296 34359869440

Peter Kagey, Illustration of T(3,3)=256
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv: 2311.13072 [math.CO], 2023.

Cf. A368220, A368222, A368223, A117401.

A368224[n_, m_] := 2^(n*m/2 - 1) (2^(n*m/2) + Boole[EvenQ[n*m]])

cp's OEIS Frontend

A176627 Triangle T(n, k) = 12^(k*(n-k)), read by rows.

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A176639 Triangle T(n, k) = 15^(k*(n-k)), read by rows.

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A176642 Triangle T(n, k) = 8^(k*(n-k)), read by rows.

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A176631 Triangle T(n, k) = 22^(k*(n-k)), read by rows.

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A176641 Triangle T(n, k) = 28^(k*(n-k)), read by rows.

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Magma

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A176643 Triangle T(n, k) = 21^(k*(n-k)), read by rows.

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A344110 Triangle read by rows: T(n,k) = 2^(n*k), n >= 0, 0 <= k <= n.