A118190 -id:A118190 - cp's OEIS frontend

A118191 Row sums of triangle A118190: a(n) = Sum_{k=0..n} 5^(k*(n-k)) for n>=0.

1, 2, 7, 52, 877, 32502, 2740627, 507843752, 214111484377, 198376465625002, 418186492923828127, 1937270172119160156252, 20419262349796295263671877, 472966350615029335022460937502, 24925857360591180741786959228515627

Offset: 0

Paul D. Hanna, Apr 15 2006

nonn

Self-convolution of A118195; in general, sqrt(Sum_{n>=0} x^n/(1-q^n*x)) is an integer series whenever q == 1 (mod 4). Also equals column 0 of the matrix square of triangle A118190, where [A118190^2](n,k) = a(n-k)*5^(k*(n-k)) for n>=k>=0.

			A(x) = 1/(1-x) + x/(1-5*x) + x^2/(1-25*x) + x^3/(1-125*x) + ...
  = 1 + 2*x + 7*x^2 + 52*x^3 + 877*x^4 + 32502*x^5 + ...

G. C. Greubel, Table of n, a(n) for n = 0..70

Cf. A118190 (triangle), A118192 (antidiagonal sums), A118195 (A(x)^(1/2)).

[(&+[5^(k*(n-k)): k in [0..n]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021

Table[Sum[5^(k*(n-k)), {k,0,n}], {n,0,30}] (* G. C. Greubel, Jun 29 2021 *)

PARI
```
a(n)=sum(k=0, n, (5^k)^(n-k))
```

[sum(5^(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/(1-5^n*x).

a(n) ~ c * 5^(n^2/4), where c = EllipticTheta[3, 0, 1/5] (in Mathematica) = JacobiTheta3(0,1/5) (in Maple) = 1.40320102401310720671088653743895... if n is even and c = EllipticTheta[2, 0, 1/5] = JacobiTheta2(0,1/5) = 1.39106543858832939481476315485543... if n is odd. - Vaclav Kotesovec, Aug 20 2025

A118192 Antidiagonal sums of triangle A118190: a(n) = Sum_{k=0..floor(n/2)} 5^(k(n-2k)) for n>=0.

Original entry on oeis.org

1, 1, 2, 6, 27, 151, 1252, 18876, 421877, 11797501, 489062502, 36867190626, 4119892578127, 576049853531251, 119400024902343752, 45003894807128984376, 25145828723919677734377, 17579646409034759521875001

Offset: 0

Paul D. Hanna, Apr 15 2006

nonn

			A(x) = 1/(1-x^2) + x/(1-5*x^2) + x^2/(1-25*x^2) + x^3/(1-125*x^2) + ...
  = 1 + x + 2*x^2 + 6*x^3 + 27*x^4 + 151*x^5 + ...

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

Cf. A118190 (triangle), A118191 (row sums).

[(&+[5^(k*(n-2*k)): k in [0..Floor(n/2)]]): n in [0..30]]; // G. C. Greubel, Jun 29 2021

Table[Sum[5^(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, (5^k)^(n-2*k) )
```

[sum(5^(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/(1-5^n*x^2).
a(2*n) = Sum_{k=0..n} 5^(2*k*(n-k)).
a(2*n+1) = Sum_{k=0..n} 5^(k*(2*(n-k)+1)).

A118193 Column 0 of the matrix inverse of triangle A118190(n,k) = 5^(k*(n-k)).

Original entry on oeis.org

1, -1, 4, -76, 7124, -3326876, 7760553124, -90490361296876, 5275336666748203124, -1537656615631182860546876, 2240970675863910673065189453124, -16329855533286908545970966339091796876, 594974481262862479448134839533519744970703124

Offset: 0

Paul D. Hanna, Apr 15 2006

sign

The entire matrix inverse of triangle A118190 is determined by column 0 (this sequence): [A118190^-1](n,k) = a(n-k)*5^(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/(1-5^n*x) by applying the inverse transformation: c(n) = Sum_{k=0..n} a(n-k)*b(k)*5^(k*(n-k)).

			Recurrence at n=4: 0 = a(0)*(5^0)^4 +a(1)*(5^1)^3 +a(2)*(5^2)^2 +a(3)*(5^3)^1 +a(4)*(5^4)^0 = 1*(5^0) - 1*(5^3) + 4*(5^4) - 76*(5^3) + 7124*(5^0).
The g.f. is illustrated by: 1 = 1/(1-x) - 1*x/(1-5*x) + 4*x^2/(1-25*x) - 76*x^3/(1-125*x) + 7124*x^4/(1-625*x) - 3326876*x^5/(1-3125*x) + 7760553124*x^6/(1-15625*x) +...

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A118190.

a[n_]:= a[n]= If[n<2, (-1)^n, -Sum[5^(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,(5^(c-1))^(r-c)))); return((T^-1)[n+1,1])}

@CachedFunction
def a(n): return (-1)^n if (n<2) else -sum(5^(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/(1-5^n*x).
0^n = Sum_{k=0..n} a(k)*5^(k*(n-k)) for n>=0.
a(n) = - Sum_{j=0..n-1} 5^(j*(n-j))*a(j) with a(0) = 1 and a(1) = -1. - G. C. Greubel, Jun 29 2021

A118194 Column 0 of the matrix log of triangle A118190, after term in row n is multiplied by n: a(n) = n[log(A118190)](n,0), where A118190(n,k) = 5^(k(n-k)).

Original entry on oeis.org

0, 1, -3, 53, -4871, 2262505, -5269940619, 61424345593757, -3580474937256484367, 1043606492389898678125009, -1520932783784930699920673828115, 11082945991224258678496051788222656261, -403804307486446123171767495567989349951171863

Offset: 0

Paul D. Hanna, Apr 15 2006

sign

The entire matrix log of triangle A118190 is determined by column 0 (this sequence): [log(A118190)](n,k) = a(n-k)5^(k*(n-k))/(n-k) for n>k>=0.

			Column 0 of log(A118190) = [0, 1, -3/2, 53/3, -4871/4, ...].
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 + 7*x^7 + ...
  = x/(1-5*x) -3*x^2/(1-25*x) +53*x^3/(1-125*x) -4871*x^4/(1-625*x) + 2262505*x^5/(1-3125*x) - 5269940619*x^6/(1-15625*x) + ...

G. C. Greubel, Table of n, a(n) for n = 0..50

Cf. A118190.

A118193[n_]:= A118193[n]= If[n<2, (-1)^n, -Sum[5^(j*(n-j))*A118193[j], {j, 0, n-1}]];
a[n_]:= a[n]= -Sum[5^(j*(n-j))*j*A118193[j], {j, 0, n}];
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,(5^(c-1))^(r-c))), L=sum(m=1,#T,-(T^0-T)^m/m));return(n*L[n+1,1])}

@CachedFunction
def A118193(n): return (-1)^n if (n<2) else -sum(5^(j*(n-j))*A118193(j) for j in (0..n-1))
def a(n): return (-1)*sum(5^(j*(n-j))*j*A118193(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/(1-5^n*x). By using the inverse transformation: a(n) = Sum_{k=0..n} k*A118193(n-k)*5^(k*(n-k)) for n>=0.

A118195 Self-convolution square-root of A118191, where A118191 is column 0 of the matrix square of triangle A118190 with A118190(n,k) = (5^k)^(n-k).

Original entry on oeis.org

1, 1, 3, 23, 411, 15771, 1353045, 252512065, 106798723795, 99080638950595, 208993838938550873, 968425792397232696773, 10208662119796586878979989, 236472963735267887311598074949, 12462692176683507314938059670486683

Offset: 0

Paul D. Hanna, Apr 15 2006

nonn

In general, sqrt( Sum_{n>=0} x^n/(1 - q^n*x) ) is an integer series whenever q == 1 (mod 4).

			A(x) = 1 + x + 3*x^2 + 23*x^3 + 411*x^4 + 15771*x^5 + ...
A(x)^2 = 1 + 2*x + 7*x^2 + 52*x^3 + 877*x^4 + 32502*x^5 + ...
= 1/(1-x) + x/(1-5x) + x^2/(1-25x) + x^3/(1-125x) + ...

G. C. Greubel, Table of n, a(n) for n = 0..75

Cf. A118190, A118191.

m:=30;
R:=PowerSeriesRing(Rationals(), m);
Coefficients(R!( Sqrt( (&+[x^j/(1-5^j*x): j in [0..m+2]]) ) )); // G. C. Greubel, Jun 30 2021

With[{m = 30}, CoefficientList[Series[Sqrt[Sum[x^j/(1 - 5^j*x), {j, 0, m + 2}]], {x, 0, m}], x]] (* G. C. Greubel, Jun 30 2021 *)

a(n)=polcoeff(sqrt(sum(k=0,n,sum(j=0, k, (5^j)^(k-j) )*x^k+x*O(x^n))),n)

m=30;
def A118195_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( sqrt(sum( x^j/(1-5^j*x) for j in (0..m+2))) ).list()
A118195_list(m) # G. C. Greubel, Jun 30 2021

G.f.: A(x) = sqrt( Sum_{n>=0} x^n/(1-5^n*x) ).

a(n) ~ A118191(n)/2. - Vaclav Kotesovec, Aug 20 2025

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

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

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

cp's OEIS Frontend

A118191 Row sums of triangle A118190: a(n) = Sum_{k=0..n} 5^(k*(n-k)) for n>=0.

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A118192 Antidiagonal sums of triangle A118190: a(n) = Sum_{k=0..floor(n/2)} 5^(k*(n-2*k)) for n>=0.

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A118193 Column 0 of the matrix inverse of triangle A118190(n,k) = 5^(k*(n-k)).

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A118194 Column 0 of the matrix log of triangle A118190, after term in row n is multiplied by n: a(n) = n*[log(A118190)](n,0), where A118190(n,k) = 5^(k*(n-k)).

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A118195 Self-convolution square-root of A118191, where A118191 is column 0 of the matrix square of triangle A118190 with A118190(n,k) = (5^k)^(n-k).

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

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A118192 Antidiagonal sums of triangle A118190: a(n) = Sum_{k=0..floor(n/2)} 5^(k(n-2k)) for n>=0.

A118194 Column 0 of the matrix log of triangle A118190, after term in row n is multiplied by n: a(n) = n[log(A118190)](n,0), where A118190(n,k) = 5^(k(n-k)).