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A118180
Triangle T(n, k) = 3^(k*(n-k)), read by rows.
19
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).
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
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Apr 15 2006
STATUS
approved