A118185 -id:A118185 - cp's OEIS frontend

A118186 Row sums of triangle A118185: a(n) = Sum_{k=0..n} 4^(k*(n-k)) for n>=0.

1, 2, 6, 34, 386, 8706, 395266, 35659778, 6476038146, 2336999211010, 1697654543745026, 2450521284684021762, 7120479243447937531906, 41112924905741324849774594, 477847273163370530909175414786

Offset: 0

Paul D. Hanna, Apr 15 2006

nonn

Also equals column 0 of the matrix square of triangle A118185, where [A118185^2](n,k) = a(n-k)*4^(k*(n-k)) for n >= k >= 0.

			A(x) = 1/(1-x) + x/(1-4x) + x^2/(1-16x) + x^3/(1-64x) + ...
  = 1 + 2*x + 6*x^2 + 34*x^3 + 386*x^4 + 8706*x^5 + ...
From _Paul D. Hanna_, Oct 14 2009: (Start)
Another g.f.: (1 + x/2^1 + x^2/2^4 + x^3/2^9 + x^4/2^16 + ...)^2
  = 1 + 2*x/2^1 + 6*x^2/2^4 + 34*x^3/2^9 + 386*x^4/2^16 + ... (End)

G. C. Greubel, Table of n, a(n) for n = 0..80

Cf. A118185 (triangle), A118187 (antidiagonal sums).

[(&+[4^(k*(n-k)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021

Table[Sum[4^(k*(n-k)), {k,0,n}], {n,0,30}] (* G. C. Greubel, Jun 29 2021 *)

PARI
```
a(n)=sum(k=0, n, (4^k)^(n-k) );
```

{a(n)=2^(n^2)*polcoeff(sum(m=0,n,x^m/2^(m^2)+x*O(x^n))^2,n)} \\ Paul D. Hanna, Oct 14 2009

[sum(4^(k*(n-k)) for k in (0..n)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

G.f.: A(x) = Sum_{n>=0} x^n/(1-4^n*x).

G.f.: Sum_{n>=1} a(n)*x^n/2^(n^2) = ( Sum_{n>=0} x^n/2^(n^2) )^2. - Paul D. Hanna, Oct 14 2009

A118187 Antidiagonal sums of triangle A118185: a(n) = Sum_{k=0..[n/2]} 4^(k(n-2k)) for n>=0.

Original entry on oeis.org

1, 1, 2, 5, 18, 81, 514, 5185, 73730, 1327361, 33685506, 1359217665, 77311508482, 5567355555841, 565149010231298, 91215553426898945, 20753150033413537794, 5977902509385249259521, 2427296516310194305630210

Offset: 0

Paul D. Hanna, Apr 15 2006

nonn

			A(x) = 1/(1-x^2) + x/(1-4*x^2) + x^2/(1-16*x^2) + x^3/(1-64*x^2) + ...
  = 1 + x + 2*x^2 + 5*x^3 + 18*x^4 + 81*x^5 + 514*x^6 + ...

G. C. Greubel, Table of n, a(n) for n = 0..100

Cf. A118185 (triangle), A118186 (row sums).

[(&+[4^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021

Table[Sum[4^(k*(n-2*k)), {k,0,Floor[n/2]}], {n,0,30}] (* G. C. Greubel, Jun 29 2021 *)

PARI
```
a(n)=sum(k=0, n\2, (4^k)^(n-2*k) )
```

[sum(4^(k*(n-2*k)) for k in (0..n//2)) for n in (0..30)] # G. C. Greubel, Jun 29 2021

G.f.: A(x) = Sum_{n>=0} x^n/(1-4^n*x^2).
a(2*n) = Sum_{k=0..n} 4^(2*k*(n-k)).
a(2*n+1) = Sum_{k=0..n} (4^k)^(2*(n-k)+1).

A118188 Column 0 of the matrix inverse of triangle A118185(n,k) = (4^k)^(n-k).

Original entry on oeis.org

1, -1, 3, -33, 1407, -237057, 158992383, -425715556353, 4556004503093247, -194971932801554579457, 33370662957719457037287423, -22845215336421444625717664940033, 62557106610069521429900219032249827327, -685195337175488637158242110253091749621661697

Offset: 0

Paul D. Hanna, Apr 15 2006

sign

The entire matrix inverse of triangle A118185 is determined by column 0 (this sequence): [A118185^-1](n,k) = a(n-k)*4^(k*(n-k)) for n>=k>=0. Any g.f. of the form: Sum_{k>=0} b(k)*x^k may be expressed as: Sum_{n>=0} c(n)*x^n/(1-4^n*x) by applying the inverse transformation: c(n) = Sum_{k=0..n} a(n-k)*b(k)*4^(k*(n-k)).

			Recurrence at n=4:
0 = a(0)*(4^0)^4 +a(1)*(4^1)^3 +a(2)*(4^2)^2 +a(3)*(4^3)^1 +a(4)*(4^4)^0
= 1*(4^0) - 1*(4^3) + 3*(4^4) - 33*(4^3) + 1407*(4^0).
The g.f. is illustrated by:
1 = 1/(1-x) - 1*x/(1-4*x) + 3*x^2/(1-16*x) - 33*x^3/(1-64*x) +
1407*x^4/(1-256*x) - 237057*x^5/(1-1024*x) + 158992383*x^6/(1-4096*x) +...

G. C. Greubel, Table of n, a(n) for n = 0..55

Cf. A118185 (triangle).

a[n_]:= a[n]= If[n<2, (-1)^n, -Sum[4^(j*(n-j))*a[j], {j, 0, n-1}]];
Table[a[n], {n, 0, 30}] (* G. C. Greubel, Jun 29 2021 *)

{a(n)=local(T=matrix(n+1,n+1,r,c,if(r>=c,(4^(c-1))^(r-c)))); return((T^-1)[n+1,1])}

@CachedFunction
def a(n): return (-1)^n if (n<2) else -sum(4^(j*(n-j))*a(j) for j in (0..n-1))
[a(n) for n in (0..30)] # G. C. Greubel, Jun 29 2021

G.f.: 1 = Sum_{n>=0} a(n)*x^n/(1-4^n*x).
0^n = Sum_{k=0..n} a(k)*4^(k*(n-k)) for n>=0.
G.f.: Sum_{n>=0} a(n)*x^n/2^(n^2) = 1/Sum_{n>=0} x^n/2^(n^2). - Vladeta Jovovic, Oct 14 2009
a(n) = (-1)*Sum_{j=0..n-1} 4^(j*(n-j))*a(j) with a(0) = 1, and a(1) = -1. - G. C. Greubel, Jun 29 2021
a(n) ~ (-1)^n * c * d^n * 2^(n^2), where d = 0.3264264803687260917234459979520857231773098381012486773185370058835770869... and c = 1.9166236268503386759188959219480676595889651334615223163023463792603010986... - Vaclav Kotesovec, Jun 09 2025

A118189 Column 0 of the matrix log of triangle A118185, after term in row n is multiplied by n: a(n) = n[log(A118185)](n,0), where A118185(n,k) = 4^(k(n-k)).

Original entry on oeis.org

0, 1, -2, 19, -764, 125701, -83499002, 222705979399, -2379643407695864, 101770765968904486921, -17414214749792087566712822, 11920352399707142353576549941259, -32640155138015817553201240150152052724, 357505372216293786145503061380504161718632461

Offset: 0

Paul D. Hanna, Apr 15 2006

sign

The entire matrix log of triangle A118185 is determined by column 0 (this sequence): [log(A118185)](n,k) = a(n-k)*4^(k*(n-k))/(n-k) for n > k >= 0.

			Column 0 of log(A118185) = [0, 1, -2/2, 19/3, -764/4, 125701/5, ...].
The g.f. is illustrated by:
x/(1-x)^2 = x + 2*x^2 + 3*x^3 + 4*x^4 + 5*x^5 + 6*x^6 + ...
  = x/(1-4*x) - 2*x^2/(1-16*x) + 19*x^3/(1-64*x) - 764*x^4/(1-256*x) + 125701*x^5/(1-1024*x) - 83499002*x^6/(1-4096*x) + 222705979399*x^7/(1-16384*x) + ...
From _Paul D. Hanna_, Oct 14 2009: (Start)
Illustrate the logarithmic g.f. by:
L(x) = x/2^1 - 2*x^2/(2*2^4) + 19*x^3/(3*2^9) - 764*x^4/(4*2^16) +- ...
where exp(L(x)) = 1 + x/2^1 + x^2/2^4 + x^3/2^9 + x^4/2^16 + ... (End)

G. C. Greubel, Table of n, a(n) for n = 0..55

Cf. A118185 (triangle), A118188.

A118188[n_]:= A118188[n]= If[n<2, (-1)^n, -Sum[4^(j*(n-j))*A118188[j], {j,0,n-1}]];
a[n_]:= a[n]= -Sum[4^(j*(n-j))*j*A118188[j], {j, 0, n}];
Table[a[n], {n, 0, 15}] (* G. C. Greubel, Jun 29 2021 *)

{a(n)=local(T=matrix(n+1,n+1,r,c,if(r>=c,(4^(c-1))^(r-c))), L=sum(m=1,#T,-(T^0-T)^m/m));return(n*L[n+1,1])}

{a(n)=n*2^(n^2)*polcoeff(log(sum(m=0,n,x^m/2^(m^2))+x*O(x^n)),n)} \\ Paul D. Hanna, Oct 14 2009

@CachedFunction
def A118188(n): return (-1)^n if (n<2) else -sum(4^(j*(n-j))*A118188(j) for j in (0..n-1))
def a(n): return (-1)*sum(4^(j*(n-j))*j*A118188(j) for j in (0..n))
[a(n) for n in (0..30)] # G. C. Greubel, Jun 29 2021

G.f.: x/(1-x)^2 = Sum_{n>=0} a(n)*x^n/(1-4^n*x).
By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118188(n-k)*4^(k*(n-k)) for n>=0.
a(2^n) is divisible by 2^n.
L.g.f.: Sum_{n>=1} a(n)*x^n/[n*2^(n^2)] = log( Sum_{n>=0} x^n/2^(n^2) ). - Paul D. Hanna, Oct 14 2009

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

Original entry on oeis.org

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

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

cp's OEIS Frontend

A118186 Row sums of triangle A118185: a(n) = Sum_{k=0..n} 4^(k*(n-k)) for n>=0.

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A118187 Antidiagonal sums of triangle A118185: a(n) = Sum_{k=0..[n/2]} 4^(k*(n-2*k)) for n>=0.

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A118188 Column 0 of the matrix inverse of triangle A118185(n,k) = (4^k)^(n-k).

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A118189 Column 0 of the matrix log of triangle A118185, after term in row n is multiplied by n: a(n) = n*[log(A118185)](n,0), where A118185(n,k) = 4^(k*(n-k)).

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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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A118187 Antidiagonal sums of triangle A118185: a(n) = Sum_{k=0..[n/2]} 4^(k(n-2k)) for n>=0.

A118189 Column 0 of the matrix log of triangle A118185, after term in row n is multiplied by n: a(n) = n[log(A118185)](n,0), where A118185(n,k) = 4^(k(n-k)).