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A257606 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) = x + 4.

%I A257606 #31 Mar 25 2022 02:18:24
%S A257606 1,4,4,16,40,16,64,296,296,64,256,1928,3552,1928,256,1024,11688,34808,
%T A257606 34808,11688,1024,4096,67656,302352,487312,302352,67656,4096,16384,
%U A257606 379240,2423016,5830000,5830000,2423016,379240,16384,65536,2076424,18330496,62617144,93280000,62617144,18330496,2076424,65536
%N A257606 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) = x + 4.
%H A257606 G. C. Greubel, <a href="/A257606/b257606.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257606 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) = x + 4.
%F A257606 Sum_{k=0..n} T(n, k) = A049388(n).
%F A257606 T(n,0) = T(n,n) = 4^n. - _Georg Fischer_, Oct 02 2021
%F A257606 From _G. C. Greubel_, Mar 24 2022: (Start)
%F A257606 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 = 1, and b = 4.
%F A257606 T(n, n-k) = T(n, k).
%F A257606 T(n, 1) = 8*5^n - 4^n*(8+n).
%F A257606 T(n, 2) = 2*((56 +15*n +n^2)*4^(n-1) - 4*(8+n)*5^n + 3*6^(n+1)). (End)
%e A257606 Triangle begins as:
%e A257606       1;
%e A257606       4,      4;
%e A257606      16,     40,      16;
%e A257606      64,    296,     296,      64;
%e A257606     256,   1928,    3552,    1928,     256;
%e A257606    1024,  11688,   34808,   34808,   11688,    1024;
%e A257606    4096,  67656,  302352,  487312,  302352,   67656,   4096;
%e A257606   16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 16384;
%t A257606 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 A257606 Table[T[n,k,1,4], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 24 2022 *)
%o A257606 (Sage)
%o A257606 def T(n,k,a,b): # A257606
%o A257606     if (k<0 or k>n): return 0
%o A257606     elif (n==0): return 1
%o A257606     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257606 flatten([[T(n,k,1,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 24 2022
%Y A257606 Cf. A008292, A049388 (row sums), A256890, A257180, A257607.
%Y A257606 Cf. A257613, A257622.
%Y A257606 Similar sequences listed in A256890.
%K A257606 nonn,tabl
%O A257606 0,2
%A A257606 _Dale Gerdemann_, May 03 2015
%E A257606 a(3) corrected by _Georg Fischer_, Oct 02 2021