A176639 -id:A176639 - cp's OEIS frontend

A117401 Triangle T(n,k) = 2^(k*(n-k)), read by rows.

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 16, 8, 1, 1, 16, 64, 64, 16, 1, 1, 32, 256, 512, 256, 32, 1, 1, 64, 1024, 4096, 4096, 1024, 64, 1, 1, 128, 4096, 32768, 65536, 32768, 4096, 128, 1, 1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1

Offset: 0

Paul D. Hanna, Mar 12 2006

Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character.

			A(x,y) = 1/(1-xy) + x/(1-2xy) + x^2/(1-4xy) + x^3/(1-8xy) + ...
Triangle begins:
  1;
  1,   1;
  1,   2,     1;
  1,   4,     4,      1;
  1,   8,    16,      8,       1;
  1,  16,    64,     64,      16,       1;
  1,  32,   256,    512,     256,      32,      1;
  1,  64,  1024,   4096,    4096,    1024,     64,     1;
  1, 128,  4096,  32768,   65536,   32768,   4096,   128,   1;
  1, 256, 16384, 262144, 1048576, 1048576, 262144, 16384, 256, 1;

G. C. Greubel, Rows n = 0..50 of the triangle, flattened
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.

Cf. A117402 (row sums), A117403 (antidiagonal sums), A002416 (central terms).

Cf. this sequence (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).

A117401:= func< n, k, m | (m+2)^(k*(n-k)) >;
[A117401(n, k, 0): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021

Table[2^((n-k)k),{n,0,10},{k,0,n}]//Flatten (* Harvey P. Dale, Jan 09 2017 *)

PARI
```
T(n,k)=if(n
				
```

def A117401(n, k, m): return (m+2)^(k*(n-k))
flatten([[A117401(n, k, 0) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

G.f.: A(x,y) = Sum_{n>=0} x^n/(1 - 2^n*x*y).
G.f. satisfies: A(x,y) = 1/(1 - x*y) + x*A(x,2*y).
Equals ConvOffsStoT transform of the 2^n series: (1, 2, 4, 8, ...); e.g., ConvOffs transform of (1, 2, 4, 8) = (1, 8, 16, 8, 1). - Gary W. Adamson, Apr 21 2008
T(n,k) = (1/n)*( 2^(n-k)*k*T(n-1,k-1) + 2^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014
Let E(x) = Sum_{n>=0} x^n/2^C(n,2).  Then E(x)*E(y*x) = Sum_{n>=0} Sum_{k=0..n} T(n,k)*y^k*x^n/2^C(n,2). - Geoffrey Critzer, May 31 2020
T(n, k, m) = (m+2)^(k*(n-k)) with m = 0. - G. C. Greubel, Jun 28 2021

A118180 Triangle T(n, k) = 3^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 9, 9, 1, 1, 27, 81, 27, 1, 1, 81, 729, 729, 81, 1, 1, 243, 6561, 19683, 6561, 243, 1, 1, 729, 59049, 531441, 531441, 59049, 729, 1, 1, 2187, 531441, 14348907, 43046721, 14348907, 531441, 2187, 1, 1, 6561, 4782969, 387420489, 3486784401, 3486784401, 387420489, 4782969, 6561, 1

Offset: 0

Paul D. Hanna, Apr 15 2006

For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-3^n*x).

			A(x,y) = 1/(1-xy) + x/(1-3xy) + x^2/(1-9xy) + x^3/(1-27xy) + ...
Triangle begins:
  1;
  1,    1;
  1,    3,      1;
  1,    9,      9,        1;
  1,   27,     81,       27,        1;
  1,   81,    729,      729,       81,        1;
  1,  243,   6561,    19683,     6561,      243,      1;
  1,  729,  59049,   531441,   531441,    59049,    729,    1;
  1, 2187, 531441, 14348907, 43046721, 14348907, 531441, 2187, 1; ...
The matrix inverse T^-1 starts:
      1;
     -1,     1;
      2,    -3,     1;
    -10,    18,    -9,    1;
    134,  -270,   162,  -27,   1;
  -4942, 10854, -7290, 1458, -81, 1; ...
where [T^-1](n,k) = A118183(n-k)*(3^k)^(n-k).

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

Cf. A118181 (row sums), A118182 (antidiagonal sums), A118183, A118184.
Cf. A117401 = ConvOffsStoT transform of 2^n.
Cf. A117401 (m=0), this sequence (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).

A118180:= func< n, k, m | (m+2)^(k*(n-k)) >;
[A118180(n, k, 1): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021

seq(seq( (3^k)^(n-k), k=0..n), n=0..12);

T[n_, k_, m_]:= (m+2)^(k*(n-k)); Table[T[n,k,1], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 28 2021 *)

T(n,k) = if(k<0 || k>n, 0, 3^(k*(n-k)));

def A118180(n, k, m): return (m+2)^(k*(n-k))
flatten([[A118180(n, k, 1) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

G.f.: A(x,y) = Sum_{n>=0} x^n/(1-3^n*x*y). G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,3*y).
Equals ConvOffsStoT transform of the 3^n series: (1, 3, 9, 27, ...); e.g., ConvOffs transform of (1, 3, 9, 27) = (1, 27, 81, 27, 1). - Gary W. Adamson, Apr 21 2008
T(n,k) = (1/n)*( 3^(n-k)*k*T(n-1,k-1) + 3^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 1. - G. C. Greubel, Jun 28 2021

A118185 Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 4, 1, 1, 16, 16, 1, 1, 64, 256, 64, 1, 1, 256, 4096, 4096, 256, 1, 1, 1024, 65536, 262144, 65536, 1024, 1, 1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1, 1, 16384, 16777216, 1073741824, 4294967296, 1073741824, 16777216, 16384, 1

Offset: 0

Paul D. Hanna, Apr 15 2006

For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-4^n*x).

Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character. For example, the matrix inverse is defined by [T^-1](n,k) = A118188(n-k)*T(n,k); also, the matrix log is given by [log(T)](n,k) = A118189(n-k)*T(n,k).

			A(x,y) = 1/(1-xy) + x/(1-4xy) + x^2/(1-16xy) + x^3/(1-64xy) + ...
Triangle begins:
  1;
  1,    1;
  1,    4,       1;
  1,   16,      16,        1;
  1,   64,     256,       64,        1;
  1,  256,    4096,     4096,      256,       1;
  1, 1024,   65536,   262144,    65536,    1024,    1;
  1, 4096, 1048576, 16777216, 16777216, 1048576, 4096, 1; ...
The matrix inverse T^-1 starts:
        1;
       -1,      1;
        3,     -4,       1;
      -33,     48,     -16,     1;
     1407,  -2112,     768,   -64,    1;
  -237057, 360192, -135168, 12288, -256, 1; ...
where [T^-1](n,k) = A118188(n-k)*4^(k*(n-k)).

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

Cf. A118186 (row sums), A118187 (antidiagonal sums), A118188, A118189.
Cf. A117401 (m=0), A118180 (m=1), this sequence (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).
T(2n,n) gives A060757.

[4^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 29 2021

Table[4^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 29 2021 *)

PARI
```
T(n, k)=if(n
				
```

flatten([[4^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 29 2021

G.f.: A(x,y) = Sum_{n>=0} x^n/(1-4^n*x*y).
G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,4*y).
T(n,k) = (1/n)*( 4^(n-k)*k*T(n-1,k-1) + 4^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 20 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 2. - G. C. Greubel, Jun 29 2021

A118190 Triangle T(n,k) = 5^(k*(n-k)) for n >= k >= 0, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 5, 1, 1, 25, 25, 1, 1, 125, 625, 125, 1, 1, 625, 15625, 15625, 625, 1, 1, 3125, 390625, 1953125, 390625, 3125, 1, 1, 15625, 9765625, 244140625, 244140625, 9765625, 15625, 1, 1, 78125, 244140625, 30517578125, 152587890625, 30517578125, 244140625, 78125, 1

Offset: 0

Paul D. Hanna, Apr 15 2006

Matrix power T^m satisfies: [T^m](n,k) = [T^m](n-k,0)*T(n,k) for all m and so the triangle has an invariant character. For example, the matrix inverse is defined by [T^-1](n,k) = A118193(n-k)*T(n,k); also, the matrix log is given by [log(T)](n,k) = A118194(n-k)*T(n,k).

For any column vector C, the matrix product of T*C transforms the g.f. of C: Sum_{n>=0} c(n)*x^n into the g.f.: Sum_{n>=0} c(n)*x^n/(1-5^n*x).

			A(x,y) = 1/(1-x*y) + x/(1-5*x*y) + x^2/(1-25*x*y) + x^3/(1-125*x*y) + ...
Triangle begins:
  1;
  1,     1;
  1,     5,       1;
  1,    25,      25,         1;
  1,   125,     625,       125,         1;
  1,   625,   15625,     15625,       625,       1;
  1,  3125,  390625,   1953125,    390625,    3125,     1;
  1, 15625, 9765625, 244140625, 244140625, 9765625, 15625, 1; ...
The matrix inverse T^-1 starts:
         1;
        -1,       1;
         4,      -5,        1;
       -76,     100,      -25,     1;
      7124,   -9500,     2500,  -125,    1;
  -3326876, 4452500, -1187500, 62500, -625, 1; ...
where [T^-1](n,k) = A118193(n-k)*(5^k)^(n-k).

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

Cf. A118191 (row sums), A118192 (antidiagonal sums), A118193, A118194.

Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), this sequence (m=3), A158116 (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15).

[5^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 29 2021

With[{m=3}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 29 2021 *)

PARI
```
T(n, k)=if(n
				
```

flatten([[5^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 29 2021

G.f.: A(x,y) = Sum_{n>=0} x^n/(1-5^n*x*y).
G.f. satisfies: A(x,y) = 1/(1-x*y) + x*A(x,5*y).
T(n,k) = (1/n)*( 5^(n-k)*k*T(n-1,k-1) + 5^k*(n-k)*T(n-1,k) ), where T(i,j)=0 if j>i. - Tom Edgar, Feb 21 2014
T(n, k, m) = (m+2)^(k*(n-k)) with m = 3. - G. C. Greubel, Jun 29 2021

A156581 Triangle T(n, k, m) = (m+2)^(k*(n-k)) with m = 15, read by rows.

Original entry on oeis.org

1, 1, 1, 1, 17, 1, 1, 289, 289, 1, 1, 4913, 83521, 4913, 1, 1, 83521, 24137569, 24137569, 83521, 1, 1, 1419857, 6975757441, 118587876497, 6975757441, 1419857, 1, 1, 24137569, 2015993900449, 582622237229761, 582622237229761, 2015993900449, 24137569, 1

Offset: 0

Roger L. Bagula, Feb 10 2009

			Triangle begins as:
  1;
  1,       1;
  1,      17,          1;
  1,     289,        289,            1;
  1,    4913,      83521,         4913,          1;
  1,   83521,   24137569,     24137569,      83521,       1;
  1, 1419857, 6975757441, 118587876497, 6975757441, 1419857, 1;

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

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), this sequence (m=15).

A156581:= func< n,k,m | (m+2)^(k*(n-k)) >;
[A156581(n,k,15): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 28 2021

(* First program *)
b[n_, k_]:= b[n, k]= If[k==0, n!, Product[Sum[Binomial[j-1, i]*(k+1)^i, {i, 0, j-1}], {j, n}]];
T[n_, k_, m_]:= T[n, k, m]= b[n, m]/(b[k, m]*b[n-k, m]);
Table[T[n, k, 15], {n,0,12}, {k,0,n}]//Flatten (* modified by G. C. Greubel, Jun 28 2021 *)
(* Second program *)
T[n_, k_, m_]:= (m+2)^(k*(n-k)); Table[T[n,k,15], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Jun 28 2021 *)

def A156581(n,k,m): return (m+2)^(k*(n-k))
flatten([[A156581(n,k,15) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 28 2021

T(n, k, m) = b(n, m)/(b(k, m)*b(n-k, m)) with b(n, k) = Product_{j=1..n} ( Sum_{i=0..j-1} binomial(j-1, i)*(k+1)^i ), b(n, 0) = n!, and m = 15.

T(n, k, m) = (m+2)^(k*(n-k)) with m = 15. - G. C. Greubel, Jun 28 2021

Edited by G. C. Greubel, Jun 28 2021

A158116 Triangle T(n,k) = 6^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 6, 1, 1, 36, 36, 1, 1, 216, 1296, 216, 1, 1, 1296, 46656, 46656, 1296, 1, 1, 7776, 1679616, 10077696, 1679616, 7776, 1, 1, 46656, 60466176, 2176782336, 2176782336, 60466176, 46656, 1, 1, 279936, 2176782336, 470184984576, 2821109907456, 470184984576, 2176782336, 279936, 1

Offset: 0

Roger L. Bagula, Mar 12 2009

			Triangle starts:
  1;
  1,     1;
  1,     6,        1;
  1,    36,       36,          1;
  1,   216,     1296,        216,          1;
  1,  1296,    46656,      46656,       1296,        1;
  1,  7776,  1679616,   10077696,    1679616,     7776,     1;
  1, 46656, 60466176, 2176782336, 2176782336, 60466176, 46656, 1;

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

Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), this sequence (m=4), A176642 (m=6), A158117 (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

Cf. this sequence (q=2), A176639 (q=3), A176641 (q=4).

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

With[{m=4}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

T(n,k) = 6^(k*(n-k));
for (n=0,11,for (k=0,n, print1(T(n,k),", "));print();); \\ Joerg Arndt, Feb 21 2014

flatten([[6^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 30 2021

T(n,k) = 6^(k*(n-k)). - Tom Edgar, Feb 20 2014
T(n,k) = (1/n)*(6^(n-k)*k*T(n-1,k-1) + 6^k*(n-k)*T(n-1,k)). - Tom Edgar, Feb 20 2014
From G. C. Greubel, Jun 30 2021: (Start)
T(n, k, m) = (m+2)^(k*(n-k)) with m = 4.
T(n, k, q) = binomial(2*q, 2)^(k*(n-k)) with q = 2. (End)

Overall edit and new name by Tom Edgar and Joerg Arndt, Feb 21 2014

A158117 Triangle T(n, k) = 10^(k*(n-k)), read by rows.

Original entry on oeis.org

1, 1, 1, 1, 10, 1, 1, 100, 100, 1, 1, 1000, 10000, 1000, 1, 1, 10000, 1000000, 1000000, 10000, 1, 1, 100000, 100000000, 1000000000, 100000000, 100000, 1, 1, 1000000, 10000000000, 1000000000000, 1000000000000, 10000000000, 1000000, 1

Offset: 0

Roger L. Bagula, Mar 12 2009

			Triangle begins as:
  1;
  1,       1;
  1,      10,           1;
  1,     100,         100,             1;
  1,    1000,       10000,          1000,             1;
  1,   10000,     1000000,       1000000,         10000,           1;
  1,  100000,   100000000,    1000000000,     100000000,      100000,       1;
  1, 1000000, 10000000000, 1000000000000, 1000000000000, 10000000000, 1000000, 1;

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

Cf. A007318 (q=0), A118180 (q=1), A158116 (q=2), this sequence (q=3), A176639 (q=4), A176643 (q=5), A176641 (q=6).

Cf. A117401 (m=0), A118180 (m=1), A118185 (m=2), A118190 (m=3), A158116 (m=4), A176642 (m=6), this sequence (m=8), A176627 (m=10), A176639 (m=13), A156581 (m=15), A176643 (m=19), A176631 (m=20), A176641 (m=26).

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

(* First program *)
T[n_, k_, q_]= Binomial[q+2,2](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=8}, Table[(m+2)^(k*(n-k)), {n,0,12}, {k,0,n}]//Flatten] (* G. C. Greubel, Jun 30 2021 *)

flatten([[10^(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) = binomial(q+2, 2)^binomial(n+1, 2), c(n, 0) = n!, and q = 3.
T(n, k, q) = binomial(q+2, 2)^(k*(n-k)) with q = 3.
T(n, k, m) = (m+2)^(k*(n-k)) with m = 8. - G. C. Greubel, Jun 30 2021

Edited by G. C. Greubel, Jun 30 2021

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A117401 Triangle T(n,k) = 2^(k*(n-k)), read by rows.

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A118180 Triangle T(n, k) = 3^(k*(n-k)), read by rows.

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A118185 Triangle T(n,k) = 4^(k*(n-k)) for n>=k>=0, read by rows.

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A118190 Triangle T(n,k) = 5^(k*(n-k)) for n >= k >= 0, read by rows.

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A156581 Triangle T(n, k, m) = (m+2)^(k*(n-k)) with m = 15, read by rows.

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A158116 Triangle T(n,k) = 6^(k*(n-k)), read by rows.

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