This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A118180 #23 Sep 08 2022 08:45:24
%S A118180 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,
%T A118180 6561,243,1,1,729,59049,531441,531441,59049,729,1,1,2187,531441,
%U A118180 14348907,43046721,14348907,531441,2187,1,1,6561,4782969,387420489,3486784401,3486784401,387420489,4782969,6561,1
%N A118180 Triangle T(n, k) = 3^(k*(n-k)), read by rows.
%C A118180 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).
%H A118180 G. C. Greubel, <a href="/A118180/b118180.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A118180 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).
%F A118180 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
%F A118180 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
%F A118180 T(n, k, m) = (m+2)^(k*(n-k)) with m = 1. - _G. C. Greubel_, Jun 28 2021
%e A118180 A(x,y) = 1/(1-xy) + x/(1-3xy) + x^2/(1-9xy) + x^3/(1-27xy) + ...
%e A118180 Triangle begins:
%e A118180 1;
%e A118180 1, 1;
%e A118180 1, 3, 1;
%e A118180 1, 9, 9, 1;
%e A118180 1, 27, 81, 27, 1;
%e A118180 1, 81, 729, 729, 81, 1;
%e A118180 1, 243, 6561, 19683, 6561, 243, 1;
%e A118180 1, 729, 59049, 531441, 531441, 59049, 729, 1;
%e A118180 1, 2187, 531441, 14348907, 43046721, 14348907, 531441, 2187, 1; ...
%e A118180 The matrix inverse T^-1 starts:
%e A118180 1;
%e A118180 -1, 1;
%e A118180 2, -3, 1;
%e A118180 -10, 18, -9, 1;
%e A118180 134, -270, 162, -27, 1;
%e A118180 -4942, 10854, -7290, 1458, -81, 1; ...
%e A118180 where [T^-1](n,k) = A118183(n-k)*(3^k)^(n-k).
%p A118180 seq(seq( (3^k)^(n-k), k=0..n), n=0..12);
%t A118180 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 *)
%o A118180 (PARI) T(n,k) = if(k<0 || k>n, 0, 3^(k*(n-k)));
%o A118180 (Magma)
%o A118180 A118180:= func< n, k, m | (m+2)^(k*(n-k)) >;
%o A118180 [A118180(n, k, 1): k in [0..n], n in [0..12]]; // _G. C. Greubel_, Jun 28 2021
%o A118180 (Sage)
%o A118180 def A118180(n, k, m): return (m+2)^(k*(n-k))
%o A118180 flatten([[A118180(n, k, 1) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Jun 28 2021
%Y A118180 Cf. A118181 (row sums), A118182 (antidiagonal sums), A118183, A118184.
%Y A118180 Cf. A117401 = ConvOffsStoT transform of 2^n.
%Y A118180 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).
%K A118180 nonn,tabl
%O A118180 0,5
%A A118180 _Paul D. Hanna_, Apr 15 2006