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

0,5

Comments

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).

Links

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

Formula

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

Example

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).

Mathematica

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

Prog

(PARI) T(n, k)=if(n<k || k<0, 0, (5^k)^(n-k) )
(Magma) [5^(k*(n-k)): k in [0..n], n in [0..12]]; // G. C. Greubel, Jun 29 2021
(Sage) flatten([[5^(k*(n-k)) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Jun 29 2021

Crossrefs

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).

Sequence in context: A152572 A203346 A176793 * A172342 A143213 A172377

Adjacent sequences: A118187 A118188 A118189 * A118191 A118192 A118193

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 15 2006

Status

approved