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A257618 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) = 8*x + 2.

%I A257618 #22 Mar 24 2022 03:31:15
%S A257618 1,2,2,4,40,4,8,472,472,8,16,4928,16992,4928,16,32,49824,433984,
%T A257618 433984,49824,32,64,499584,9505728,22567168,9505728,499584,64,128,
%U A257618 4999040,192085632,909941120,909941120,192085632,4999040,128
%N A257618 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) = 8*x + 2.
%H A257618 G. C. Greubel, <a href="/A257618/b257618.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257618 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) = 8*x + 2.
%F A257618 Sum_{k=0..n} T(n, k) = A144828(n).
%F A257618 From _G. C. Greubel_, Mar 24 2022: (Start)
%F A257618 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 = 8, and b = 2.
%F A257618 T(n, n-k) = T(n, k).
%F A257618 T(n, 0) = A000079(n).
%F A257618 T(n, 1) = 2^(n-1)*(5^n - 2*n - 1).
%F A257618 T(n, 2) = 2^(n-3)*(3^(2*n+1) -2*(2*n+1)*5^n -1 +4*n^2). (End)
%e A257618 Triangle begins as:
%e A257618     1;
%e A257618     2,       2;
%e A257618     4,      40,         4;
%e A257618     8,     472,       472,         8;
%e A257618    16,    4928,     16992,      4928,        16;
%e A257618    32,   49824,    433984,    433984,     49824,        32;
%e A257618    64,  499584,   9505728,  22567168,   9505728,    499584,      64;
%e A257618   128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128;
%t A257618 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 A257618 Table[T[n,k,8,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 24 2022 *)
%o A257618 (Sage)
%o A257618 def T(n,k,a,b): # A257618
%o A257618     if (k<0 or k>n): return 0
%o A257618     elif (n==0): return 1
%o A257618     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257618 flatten([[T(n,k,8,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 24 2022
%Y A257618 Cf. A000079, A144828 (row sums), A167884.
%Y A257618 Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257619.
%Y A257618 Similar sequences listed in A256890.
%K A257618 nonn,tabl
%O A257618 0,2
%A A257618 _Dale Gerdemann_, May 09 2015