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

Author

Paul D. Hanna, Apr 15 2006

Keywords

• nonn
• tabl

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).

Examples

			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).
		

Links

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

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).

Programs

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

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 *)

PARI

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

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

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