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A257626 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.

%I A257626 #15 Mar 21 2022 02:14:32
%S A257626 1,6,6,36,108,36,216,1404,1404,216,1296,15876,33696,15876,1296,7776,
%T A257626 166212,642492,642492,166212,7776,46656,1659204,10701720,19274760,
%U A257626 10701720,1659204,46656,279936,16052580,163263924,481752360,481752360,163263924,16052580,279936
%N A257626 Triangle read by rows: T(n,k) = t(n-k, k); t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1), where f(x) = 3*x + 6.
%H A257626 G. C. Greubel, <a href="/A257626/b257626.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257626 T(n,k) = t(n-k, k); 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), where f(x) = 3*x + 6.
%F A257626 Sum_{k=0..n} T(n, k) = A051609(n).
%F A257626 T(n, k) = (a*k + b)*T(n-1, k) + (a*(n-k) + b)*T(n-1, k-1), with T(n, 0) = 1, a = 3, and b = 6. - _G. C. Greubel_, Mar 20 2022
%e A257626 Triangle begins as:
%e A257626        1;
%e A257626        6,        6;
%e A257626       36,      108,        36;
%e A257626      216,     1404,      1404,       216;
%e A257626     1296,    15876,     33696,     15876,      1296;
%e A257626     7776,   166212,    642492,    642492,    166212,      7776;
%e A257626    46656,  1659204,  10701720,  19274760,  10701720,   1659204,    46656;
%e A257626   279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936;
%t A257626 T[n_, k_, a_, b_]:= T[n, k, a, b]= If[k<0 || k>n, 0, If[n==0, 1, (a*(n-k)+b)*T[n-1, k-1, a, b] + (a*k+b)*T[n-1, k, a, b]]];
%t A257626 Table[T[n,k,3,6], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)
%o A257626 (Sage)
%o A257626 def T(n,k,a,b): # A257626
%o A257626     if (k<0 or k>n): return 0
%o A257626     elif (n==0): return 1
%o A257626     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257626 flatten([[T(n,k,3,6) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022
%Y A257626 Cf. A051609 (row sums), A142458, A257610, A257620, A257622, A257624.
%Y A257626 See similar sequences listed in A256890.
%K A257626 nonn,tabl
%O A257626 0,2
%A A257626 _Dale Gerdemann_, May 10 2015