cp's OEIS Frontend

A257622 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.

%I A257622 #13 Mar 21 2022 02:14:37
%S A257622 1,4,4,16,56,16,64,552,552,64,256,4696,11040,4696,256,1024,36968,
%T A257622 171448,171448,36968,1024,4096,278232,2305968,4457648,2305968,278232,
%U A257622 4096,16384,2037736,28346088,94844912,94844912,28346088,2037736,16384
%N A257622 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 A257622 G. C. Greubel, <a href="/A257622/b257622.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257622 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 A257622 Sum_{k=0..n} T(n, k) = A051605(n).
%F A257622 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 A257622 Triangle begins as:
%e A257622       1;
%e A257622       4,       4;
%e A257622      16,      56,       16;
%e A257622      64,     552,      552,       64;
%e A257622     256,    4696,    11040,     4696,      256;
%e A257622    1024,   36968,   171448,   171448,    36968,     1024;
%e A257622    4096,  278232,  2305968,  4457648,  2305968,   278232,    4096;
%e A257622   16384, 2037736, 28346088, 94844912, 94844912, 28346088, 2037736, 16384;
%t A257622 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 A257622 Table[T[n,k,3,4], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)
%o A257622 (Sage)
%o A257622 def T(n,k,a,b): # A257622
%o A257622     if (k<0 or k>n): return 0
%o A257622     elif (n==0): return 1
%o A257622     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257622 flatten([[T(n,k,3,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022
%Y A257622 Cf. A051605 (row sums), A142458, A257610, A257620, A257624, A257626.
%Y A257622 Cf. A257606, A257613.
%Y A257622 See similar sequences listed in A256890.
%K A257622 nonn,tabl
%O A257622 0,2
%A A257622 _Dale Gerdemann_, May 10 2015