A118192 -id:A118192 - cp's OEIS frontend

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

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

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

Original entry on oeis.org

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

cp's OEIS Frontend

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

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