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

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

0,5

Comments

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).

Links

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

Formula

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

Example

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).

Maple

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

Mathematica

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 *)

Prog

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

Crossrefs

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).

Sequence in context: A157179 A152655 A144493 * A176482 A045912 A290554

Adjacent sequences: A118177 A118178 A118179 * A118181 A118182 A118183

Keyword

nonn,tabl

Author

Paul D. Hanna, Apr 15 2006

Status

approved