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A257627 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.

%I A257627 #12 Feb 22 2022 03:41:09
%S A257627 1,3,3,9,60,9,27,753,753,27,81,8178,25602,8178,81,243,84291,631506,
%T A257627 631506,84291,243,729,852144,13348623,30312288,13348623,852144,729,
%U A257627 2187,8554245,259308063,1141302225,1141302225,259308063,8554245,2187
%N A257627 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 A257627 G. C. Greubel, <a href="/A257627/b257627.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257627 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 A257627 Sum_{k=0..n} T(n, k) = A049209(n).
%F A257627 From _G. C. Greubel_, Feb 22 2022: (Start)
%F A257627 t(k, n) = t(n, k).
%F A257627 T(n, n-k) = T(n, k).
%F A257627 t(0, n) = T(n, 0) = A000244(n). (End)
%e A257627 Array t(n, k) begins as:
%e A257627     1,       3,          9,            27,              81, ... A000244;
%e A257627     3,      60,        753,          8178,           84291, ...;
%e A257627     9,     753,      25602,        631506,        13348623, ...;
%e A257627    27,    8178,     631506,      30312288,      1141302225, ...;
%e A257627    81,   84291,   13348623,    1141302225,     70760737950, ...;
%e A257627   243,  852144,  259308063,   37244959794,   3608891348622, ...;
%e A257627   729, 8554245, 4793178096, 1109572049376, 161806374029202, ...;
%e A257627 Triangle, T(n, k) begins as:
%e A257627      1;
%e A257627      3,       3;
%e A257627      9,      60,         9;
%e A257627     27,     753,       753,         27;
%e A257627     81,    8178,     25602,       8178,         81;
%e A257627    243,   84291,    631506,     631506,      84291,       243;
%e A257627    729,  852144,  13348623,   30312288,   13348623,    852144,     729;
%e A257627   2187, 8554245, 259308063, 1141302225, 1141302225, 259308063, 8554245, 2187;
%t A257627 f[n_]:= 7*n+3;
%t A257627 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 A257627 T[n_, k_]= t[n-k, k];
%t A257627 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 22 2022 *)
%o A257627 (Sage)
%o A257627 def f(n): return 7*n+3
%o A257627 @CachedFunction
%o A257627 def t(n,k):
%o A257627     if (n<0 or k<0): return 0
%o A257627     elif (n==0 and k==0): return 1
%o A257627     else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)
%o A257627 def A257627(n,k): return t(n-k,k)
%o A257627 flatten([[A257627(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 22 2022
%Y A257627 Cf. A000244, A038221, A049209 (row sums), A142462.
%Y A257627 Cf. A257180, A257611, A257617, A257620, A257621, A257623, A257625.
%Y A257627 See similar sequences listed in A256890.
%K A257627 nonn,tabl
%O A257627 0,2
%A A257627 _Dale Gerdemann_, May 10 2015