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%I #22 Feb 22 2022 10:28:32
%S 1,3,3,9,24,9,27,141,141,27,81,726,1410,726,81,243,3471,11406,11406,
%T 3471,243,729,15828,81327,136872,81327,15828,729,2187,69873,533259,
%U 1390521,1390521,533259,69873,2187,6561,301362,3295152,12609198,19467294,12609198,3295152,301362,6561,19683,1277619,19489380,105311556,237144642,237144642,105311556,19489380,1277619,19683
%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) = x + 3.
%H G. C. Greubel, <a href="/A257180/b257180.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), and f(x) = x + 3.
%F Sum_{k=0..n} T(n, k) = A001725(n+5).
%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, 243, ... A000244;
%e 3, 24, 141, 726, 3471, 15828, ...;
%e 9, 141, 1410, 11406, 81327, 533259, ...;
%e 27, 726, 11406, 136872, 1390521, 12609198, ...;
%e 81, 3471, 81327, 1390521, 19467294, 237144642, ...;
%e 243, 15828, 533259, 12609198, 237144642, 3794314272, ...;
%e 729, 69873, 3295152, 105311556, 2607816498, 53824862658, ...;
%e Triangle T(n,k) begins as:
%e 1;
%e 3, 3;
%e 9, 24, 9;
%e 27, 141, 141, 27;
%e 81, 726, 1410, 726, 81;
%e 243, 3471, 11406, 11406, 3471, 243;
%e 729, 15828, 81327, 136872, 81327, 15828, 729;
%e 2187, 69873, 533259, 1390521, 1390521, 533259, 69873, 2187;
%e 6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561;
%t f[n_]:= 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 (PARI) f(x) = x + 3;
%o T(n, k) = t(n-k, k);
%o t(n, m) = {if (!n && !m, return(1)); if (n < 0 || m < 0, return (0)); f(m)*t(n-1,m) + f(n)*t(n,m-1);}
%o tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print(););} \\ _Michel Marcus_, Apr 23 2015
%o (Sage)
%o def f(n): return 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, A008292, A001725 (row sums), A038221, A256890, A257606.
%Y Cf. A257607, A257611, A257620, A257621, A257623, A257625, A257627.
%Y Similar sequences listed in A256890.
%K nonn,tabl
%O 0,2
%A _Dale Gerdemann_, Apr 17 2015
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