%I #12 Feb 22 2022 03:41:09
%S 1,3,3,9,60,9,27,753,753,27,81,8178,25602,8178,81,243,84291,631506,
%T 631506,84291,243,729,852144,13348623,30312288,13348623,852144,729,
%U 2187,8554245,259308063,1141302225,1141302225,259308063,8554245,2187
%N Triangle, read by rows, T(n,k) = t(n-k, k) where t(n,m) = f(m)*t(n-1,m) + f(n)*t(n,m-1) and f(x) = 7*x + 3.
%H G. C. Greubel, <a href="/A257627/b257627.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), where f(x) = 7*x + 3.
%F Sum_{k=0..n} T(n, k) = A049209(n).
%F From _G. C. Greubel_, Feb 22 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, ... A000244;
%e 3, 60, 753, 8178, 84291, ...;
%e 9, 753, 25602, 631506, 13348623, ...;
%e 27, 8178, 631506, 30312288, 1141302225, ...;
%e 81, 84291, 13348623, 1141302225, 70760737950, ...;
%e 243, 852144, 259308063, 37244959794, 3608891348622, ...;
%e 729, 8554245, 4793178096, 1109572049376, 161806374029202, ...;
%e Triangle, T(n, k) begins as:
%e 1;
%e 3, 3;
%e 9, 60, 9;
%e 27, 753, 753, 27;
%e 81, 8178, 25602, 8178, 81;
%e 243, 84291, 631506, 631506, 84291, 243;
%e 729, 852144, 13348623, 30312288, 13348623, 852144, 729;
%e 2187, 8554245, 259308063, 1141302225, 1141302225, 259308063, 8554245, 2187;
%t f[n_]:= 7*n+3;
%t t[n_, k_]:= t[n,k]= If[n<0 || k<0, 0, If[n==0 && k==0, 1, f[k]*t[n-1,k] +f[n]*t[n,k-1]]];
%t T[n_, k_]= t[n-k, k];
%t Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 22 2022 *)
%o (Sage)
%o def f(n): return 7*n+3
%o @CachedFunction
%o def t(n,k):
%o if (n<0 or k<0): return 0
%o elif (n==0 and k==0): return 1
%o else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)
%o def A257627(n,k): return t(n-k,k)
%o flatten([[A257627(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 22 2022
%Y Cf. A000244, A038221, A049209 (row sums), A142462.
%Y Cf. A257180, A257611, A257617, A257620, A257621, A257623, A257625.
%Y See similar sequences listed in A256890.
%K nonn,tabl
%O 0,2
%A _Dale Gerdemann_, May 10 2015