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A257180 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.

%I A257180 #22 Feb 22 2022 10:28:32
%S A257180 1,3,3,9,24,9,27,141,141,27,81,726,1410,726,81,243,3471,11406,11406,
%T A257180 3471,243,729,15828,81327,136872,81327,15828,729,2187,69873,533259,
%U A257180 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 A257180 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 A257180 G. C. Greubel, <a href="/A257180/b257180.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257180 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 A257180 Sum_{k=0..n} T(n, k) = A001725(n+5).
%F A257180 From _G. C. Greubel_, Feb 22 2022: (Start)
%F A257180 t(k, n) = t(n, k).
%F A257180 T(n, n-k) = T(n, k).
%F A257180 t(0, n) = T(n, 0) = A000244(n). (End)
%e A257180 Array t(n,k) begins as:
%e A257180     1,     3,       9,        27,         81,         243, ... A000244;
%e A257180     3,    24,     141,       726,       3471,       15828, ...;
%e A257180     9,   141,    1410,     11406,      81327,      533259, ...;
%e A257180    27,   726,   11406,    136872,    1390521,    12609198, ...;
%e A257180    81,  3471,   81327,   1390521,   19467294,   237144642, ...;
%e A257180   243, 15828,  533259,  12609198,  237144642,  3794314272, ...;
%e A257180   729, 69873, 3295152, 105311556, 2607816498, 53824862658, ...;
%e A257180 Triangle T(n,k) begins as:
%e A257180      1;
%e A257180      3,      3;
%e A257180      9,     24,       9;
%e A257180     27,    141,     141,       27;
%e A257180     81,    726,    1410,      726,       81;
%e A257180    243,   3471,   11406,    11406,     3471,      243;
%e A257180    729,  15828,   81327,   136872,    81327,    15828,     729;
%e A257180   2187,  69873,  533259,  1390521,  1390521,   533259,   69873,   2187;
%e A257180   6561, 301362, 3295152, 12609198, 19467294, 12609198, 3295152, 301362, 6561;
%t A257180 f[n_]:= n+3;
%t A257180 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 A257180 T[n_, k_]= t[n-k, k];
%t A257180 Table[T[n, k], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Feb 22 2022 *)
%o A257180 (PARI) f(x) = x + 3;
%o A257180 T(n, k) = t(n-k, k);
%o A257180 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 A257180 tabl(nn) = {for (n=0, nn, for (k=0, n, print1(T(n, k), ", ");); print(););} \\ _Michel Marcus_, Apr 23 2015
%o A257180 (Sage)
%o A257180 def f(n): return n+3
%o A257180 @CachedFunction
%o A257180 def t(n,k):
%o A257180     if (n<0 or k<0): return 0
%o A257180     elif (n==0 and k==0): return 1
%o A257180     else: return f(k)*t(n-1, k) + f(n)*t(n, k-1)
%o A257180 def A257627(n,k): return t(n-k,k)
%o A257180 flatten([[A257627(n,k) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Feb 22 2022
%Y A257180 Cf. A000244, A008292, A001725 (row sums), A038221, A256890, A257606.
%Y A257180 Cf. A257607, A257611, A257620, A257621, A257623, A257625, A257627.
%Y A257180 Similar sequences listed in A256890.
%K A257180 nonn,tabl
%O A257180 0,2
%A A257180 _Dale Gerdemann_, Apr 17 2015