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A257616 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) = 6*x + 2.

%I A257616 #13 Mar 22 2022 03:11:36
%S A257616 1,2,2,4,32,4,8,312,312,8,16,2656,8736,2656,16,32,21664,175424,175424,
%T A257616 21664,32,64,174336,3019200,7016960,3019200,174336,64,128,1397120,
%U A257616 47847552,218838400,218838400,47847552,1397120,128,256,11182592,722956288,5907889664,11379596800,5907889664,722956288,11182592,256
%N A257616 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) = 6*x + 2.
%H A257616 G. C. Greubel, <a href="/A257616/b257616.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257616 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) = 6*x + 2.
%F A257616 Sum_{k=0..n} T(n, k) = A049308(n).
%F A257616 From _G. C. Greubel_, Mar 21 2022: (Start)
%F A257616 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 = 6, and b = 2.
%F A257616 T(n, n-k) = T(n, k).
%F A257616 T(n, 0) = A000079(n).
%F A257616 T(n, 1) = (2^n/3)*(2^(2*n+1) - (3*n+2)). (End)
%e A257616 Triangle begins as:
%e A257616     1;
%e A257616     2,       2;
%e A257616     4,      32,        4;
%e A257616     8,     312,      312,         8;
%e A257616    16,    2656,     8736,      2656,        16;
%e A257616    32,   21664,   175424,    175424,     21664,       32;
%e A257616    64,  174336,  3019200,   7016960,   3019200,   174336,      64;
%e A257616   128, 1397120, 47847552, 218838400, 218838400, 47847552, 1397120, 128;
%t A257616 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 A257616 Table[T[n,k,6,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 21 2022 *)
%o A257616 (Sage)
%o A257616 def T(n,k,a,b): # A257610
%o A257616     if (k<0 or k>n): return 0
%o A257616     elif (n==0): return 1
%o A257616     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257616 flatten([[T(n,k,6,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 21 2022
%Y A257616 Cf. A000079, A049308 (row sums), A142461, A257625.
%Y A257616 Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257617, A257618, A257619.
%Y A257616 Similar sequences listed in A256890.
%K A257616 nonn,tabl
%O A257616 0,2
%A A257616 _Dale Gerdemann_, May 09 2015