A257606 - OEIS

%I #31 Mar 25 2022 02:18:24

%S 1,4,4,16,40,16,64,296,296,64,256,1928,3552,1928,256,1024,11688,34808,

%T 34808,11688,1024,4096,67656,302352,487312,302352,67656,4096,16384,

%U 379240,2423016,5830000,5830000,2423016,379240,16384,65536,2076424,18330496,62617144,93280000,62617144,18330496,2076424,65536

%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) = x + 4.

%H G. C. Greubel, <a href="/A257606/b257606.txt">Rows n = 0..50 of the triangle, flattened</a>

%F 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 Sum_{k=0..n} T(n, k) = A049388(n).

%F T(n,0) = T(n,n) = 4^n. - _Georg Fischer_, Oct 02 2021

%F From _G. C. Greubel_, Mar 24 2022: (Start)

%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 = 1, and b = 4.

%F T(n, n-k) = T(n, k).

%F T(n, 1) = 8*5^n - 4^n*(8+n).

%F T(n, 2) = 2*((56 +15*n +n^2)*4^(n-1) - 4*(8+n)*5^n + 3*6^(n+1)). (End)

%e Triangle begins as:

%e       1;

%e       4,      4;

%e      16,     40,      16;

%e      64,    296,     296,      64;

%e     256,   1928,    3552,    1928,     256;

%e    1024,  11688,   34808,   34808,   11688,    1024;

%e    4096,  67656,  302352,  487312,  302352,   67656,   4096;

%e   16384, 379240, 2423016, 5830000, 5830000, 2423016, 379240, 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,1,4], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 24 2022 *)

%o (Sage)

%o def T(n,k,a,b): # A257606

%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,1,4) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 24 2022

%Y Cf. A008292, A049388 (row sums), A256890, A257180, A257607.

%Y Cf. A257613, A257622.

%Y Similar sequences listed in A256890.

%K nonn,tabl

%O 0,2

%A _Dale Gerdemann_, May 03 2015

%E a(3) corrected by _Georg Fischer_, Oct 02 2021