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 A257621 #20 Mar 01 2022 05:32:28
%S A257621 1,3,3,9,42,9,27,393,393,27,81,3156,8646,3156,81,243,23631,142446,
%T A257621 142446,23631,243,729,171006,2015895,4273380,2015895,171006,729,2187,
%U A257621 1216725,26107983,102402705,102402705,26107983,1216725,2187,6561,8584872,320039388,2136524184,3891302790,2136524184,320039388,8584872,6561
%N A257621 Triangle read by rows: T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.
%H A257621 G. C. Greubel, <a href="/A257621/b257621.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257621 T(n, k) = t(n-k, k), where t(0,0) = 1, t(n,m) = 0 if n < 0 or m < 0, else t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), and f(n) = 4*n + 3.
%F A257621 Sum_{k=0..n} T(n, k) = A000407(n).
%F A257621 From _G. C. Greubel_, Mar 01 2022: (Start)
%F A257621 t(k, n) = t(n, k).
%F A257621 T(n, n-k) = T(n, k).
%F A257621 t(0, n) = T(n, 0) = A000244(n). (End)
%e A257621 Array t(n,k) begins as:
%e A257621 1, 3, 9, 27, 81, ...;
%e A257621 3, 42, 393, 3156, 23631, ...;
%e A257621 9, 393, 8646, 142446, 2015895, ...;
%e A257621 27, 3156, 142446, 4273380, 102402705, ...;
%e A257621 81, 23631, 2015895, 102402705, 3891302790, ...;
%e A257621 243, 171006, 26107983, 2136524184, 123074809242, ...;
%e A257621 729, 1216725, 320039388, 40688926236, 3437022383970, ...;
%e A257621 Triangle T(n,k) begins as:
%e A257621 1;
%e A257621 3, 3;
%e A257621 9, 42, 9;
%e A257621 27, 393, 393, 27;
%e A257621 81, 3156, 8646, 3156, 81;
%e A257621 243, 23631, 142446, 142446, 23631, 243;
%e A257621 729, 171006, 2015895, 4273380, 2015895, 171006, 729;
%e A257621 2187, 1216725, 26107983, 102402705, 102402705, 26107983, 1216725, 2187;
%t A257621 t[n_, k_, p_, q_]:= t[n, k, p, q] = If[n<0 || k<0, 0, If[n==0 && k==0, 1, (p*k+q)*t[n-1,k,p,q] + (p*n+q)*t[n,k-1,p,q]]];
%t A257621 T[n_, k_, p_, q_]= t[n-k, k, p, q];
%t A257621 Table[T[n,k,4,3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 01 2022 *)
%o A257621 (Sage)
%o A257621 @CachedFunction
%o A257621 def t(n,k,p,q):
%o A257621 if (n<0 or k<0): return 0
%o A257621 elif (n==0 and k==0): return 1
%o A257621 else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
%o A257621 def A257621(n,k): return t(n-k,k,4,3)
%o A257621 flatten([[A257621(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 01 2022
%Y A257621 Cf. A000407 (row sums), A142459, A257612.
%Y A257621 Cf. A038221, A257180, A257611, A257620, A257623, A257625, A257627.
%Y A257621 Similar sequences listed in A256890.
%K A257621 nonn,tabl
%O A257621 0,2
%A A257621 _Dale Gerdemann_, May 09 2015