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A257625 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.

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