%I #13 Mar 01 2022 05:32:42
%S 1,3,3,9,54,9,27,621,621,27,81,6156,18630,6156,81,243,57591,408726,
%T 408726,57591,243,729,526338,7685847,17166492,7685847,526338,729,2187,
%U 4765473,132656859,568014201,568014201,132656859,4765473,2187
%N 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) = 6*n + 3.
%H G. C. Greubel, <a href="/A257625/b257625.txt">Rows n = 0..50 of the triangle, flattened</a>
%F 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) = 6*n + 3.
%F Sum_{k=0..n} T(n, k) = A047058(n).
%F From _G. C. Greubel_, Mar 01 2022: (Start)
%F t(k, n) = t(n, k).
%F T(n, n-k) = T(n, k).
%F t(0, n) = T(n, 0) = A000244(n). (End)
%e Array t(n,k) begins as:
%e 1, 3, 9, 27, 81, ...;
%e 3, 54, 621, 6156, 57591, ...;
%e 9, 621, 18630, 408726, 7685847, ...;
%e 27, 6156, 408726, 17166492, 568014201, ...;
%e 81, 57591, 7685847, 568014201, 30672766854, ...;
%e 243, 526338, 132656859, 16305974568, 1366261865802, ...;
%e 729, 4765473, 2175706332, 427278012876, 53552912878818, ...;
%e Triangle T(n,k) begins as:
%e 1;
%e 3, 3;
%e 9, 54, 9;
%e 27, 621, 621, 27;
%e 81, 6156, 18630, 6156, 81;
%e 243, 57591, 408726, 408726, 57591, 243;
%e 729, 526338, 7685847, 17166492, 7685847, 526338, 729;
%e 2187, 4765473, 132656859, 568014201, 568014201, 132656859, 4765473, 2187;
%t 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 T[n_, k_, p_, q_]= t[n-k, k, p, q];
%t Table[T[n,k,6,3], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 01 2022 *)
%o (Sage)
%o @CachedFunction
%o def t(n,k,p,q):
%o if (n<0 or k<0): return 0
%o elif (n==0 and k==0): return 1
%o else: return (p*k+q)*t(n-1,k,p,q) + (p*n+q)*t(n,k-1,p,q)
%o def A257625(n,k): return t(n-k,k,6,3)
%o flatten([[A257625(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 01 2022
%Y Cf. A047058 (row sums), A142461, A257616.
%Y Cf. A038221, A257180, A257611, A257620, A257621, A257623, A257625, A257627.
%Y See similar sequences listed in A256890.
%K nonn,tabl
%O 0,2
%A _Dale Gerdemann_, May 10 2015