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