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