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.
a(n) = 3^(n-2)*n*(2+n);
a(n) = 3^(n-4)*n*(-6+20*n+12*n^2+n^3);
a(n) = 3^(n-5)*n*(-30+10*n+80*n^2+20*n^3+n^4);
f_n(m) = (1/3^(m-n)) * Sum_{k=0..m} k^n * 2^(m-k) * binomial(m,k).
f_0(m) = 1.
f_1(m) = m.
f_2(m) = 2*m + m^2.
f_3(m) = 2*m + 6*m^2 + m^3.
Triangle begins:
1;
0, 1;
0, 2, 1;
0, 2, 6, 1;
0, -6, 20, 12, 1;
0, -30, 10, 80, 20, 1;
0, 42, -320, 270, 220, 30, 1;
...
T(n, k) = sum(j=k, n, 3^(n-j)*stirling(n, j, 2)*stirling(j, k, 1));
def a_row(n):
s = sum(3^(n-k)*stirling_number2(n, k)*falling_factorial(x, k) for k in (0..n))
return expand(s).list()
for n in (0..10): print(a_row(n))