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%I #13 Mar 21 2022 02:14:37
%S 1,4,4,16,56,16,64,552,552,64,256,4696,11040,4696,256,1024,36968,
%T 171448,171448,36968,1024,4096,278232,2305968,4457648,2305968,278232,
%U 4096,16384,2037736,28346088,94844912,94844912,28346088,2037736,16384
%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 + 4.
%H G. C. Greubel, <a href="/A257622/b257622.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 + 4.
%F Sum_{k=0..n} T(n, k) = A051605(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 = 4. - _G. C. Greubel_, Mar 20 2022
%e Triangle begins as:
%e 1;
%e 4, 4;
%e 16, 56, 16;
%e 64, 552, 552, 64;
%e 256, 4696, 11040, 4696, 256;
%e 1024, 36968, 171448, 171448, 36968, 1024;
%e 4096, 278232, 2305968, 4457648, 2305968, 278232, 4096;
%e 16384, 2037736, 28346088, 94844912, 94844912, 28346088, 2037736, 16384;
%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,3,4], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)
%o (Sage)
%o def T(n,k,a,b): # A257622
%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,3,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022
%Y Cf. A051605 (row sums), A142458, A257610, A257620, A257624, A257626.
%Y Cf. A257606, A257613.
%Y See similar sequences listed in A256890.
%K nonn,tabl
%O 0,2
%A _Dale Gerdemann_, May 10 2015
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