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 A269946 #11 Jun 20 2017 09:51:59 %S A269946 1,0,1,0,2,1,0,18,18,1,0,504,648,72,1,0,32760,47160,7200,200,1,0, %T A269946 4127760,6305040,1141560,45000,450,1,0,895723920,1416456720,283704120, %U A269946 13741560,198450,882,1,0,308129028480,498072032640,106386981120,5876519040,106616160,691488,1568,1 %N A269946 Triangle read by rows, Lah numbers of order 3, T(n,n) = 1, T(n,k) = 0 if k<0 or k>n, otherwise T(n,k) = T(n-1,k-1)+((n-1)^3+k^3)*T(n-1, k), for n>=0 and 0<=k<=n. %H A269946 Peter Luschny, <a href="http://oeis.org/wiki/User:Peter_Luschny/P-Transform">The P-transform</a>. %F A269946 T(n,k) = Sum_{j=k..n} A269947(n,j)*A269948(j,k). %F A269946 T(n,1) = Product_{k=1..n} (k-1)^3+1 for n>=1 (cf. A255433). %F A269946 T(n,n-1) = (n-1)^2*n^2/2 for n>=1 (cf. A163102). %e A269946 Triangle starts: %e A269946 [1] %e A269946 [0, 1] %e A269946 [0, 2, 1] %e A269946 [0, 18, 18, 1] %e A269946 [0, 504, 648, 72, 1] %e A269946 [0, 32760, 47160, 7200, 200, 1] %e A269946 [0, 4127760, 6305040, 1141560, 45000, 450, 1] %p A269946 T := proc(n, k) option remember; %p A269946 `if`(n=k, 1, %p A269946 `if`(k<0 or k>n, 0, %p A269946 T(n-1, k-1) + ((n-1)^3+k^3) * T(n-1, k) )) end: %p A269946 for n from 0 to 6 do seq(T(n,k), k=0..n) od; %t A269946 T[n_, n_] = 1; T[_, 0] = 0; T[n_, k_] /; 0 < k < n := T[n, k] = T[n-1, k-1] + ((n-1)^3 + k^3)*T[n-1, k]; T[_, _] = 0; %t A269946 Table[T[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 20 2017 *) %Y A269946 Cf. A038207 (order 0), A111596 (order 1), A268434 (order 2). %Y A269946 Cf. A255433, A163102, A269947, A269948. %K A269946 nonn,tabl %O A269946 0,5 %A A269946 _Peter Luschny_, Mar 22 2016