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A257617 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) = 7*x + 2.

%I A257617 #20 Mar 24 2022 03:31:02
%S A257617 1,2,2,4,36,4,8,388,388,8,16,3676,12416,3676,16,32,33564,283204,
%T A257617 283204,33564,32,64,303260,5538184,13027384,5538184,303260,64,128,
%U A257617 2732156,99831564,465775352,465775352,99831564,2732156,128
%N A257617 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) = 7*x + 2.
%H A257617 G. C. Greubel, <a href="/A257617/b257617.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257617 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) = 7*x + 2.
%F A257617 Sum_{k=0..n} T(n, k) = A144827(n).
%F A257617 From _G. C. Greubel_, Mar 24 2022: (Start)
%F A257617 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 = 7, and b = 2.
%F A257617 T(n, n-k) = T(n, k).
%F A257617 T(n, 0) = A000079(n).
%F A257617 T(n, 1) = (4*9^n - 2^n*(7*n + 4))/7.
%F A257617 T(n, 2) = (2^(n-1)*(49*n^2 +7*n -12) + 11*2^(4*n+1) - 4*(7*n+4)*9^n)/49. (End)
%e A257617     1;
%e A257617     2,       2;
%e A257617     4,      36,        4;
%e A257617     8,     388,      388,         8;
%e A257617    16,    3676,    12416,      3676,        16;
%e A257617    32,   33564,   283204,    283204,     33564,       32;
%e A257617    64,  303260,  5538184,  13027384,   5538184,   303260,      64;
%e A257617   128, 2732156, 99831564, 465775352, 465775352, 99831564, 2732156, 128;
%t A257617 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 A257617 Table[T[n,k,7,2], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 24 2022 *)
%o A257617 (Sage)
%o A257617 def T(n,k,a,b): # A257617
%o A257617     if (k<0 or k>n): return 0
%o A257617     elif (n==0): return 1
%o A257617     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257617 flatten([[T(n,k,7,2) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 24 2022
%Y A257617 Cf. A000079, A142462, A144827 (row sums), A257627.
%Y A257617 Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257618, A257619
%Y A257617 Similar sequences listed in A256890.
%K A257617 nonn,tabl
%O A257617 0,2
%A A257617 _Dale Gerdemann_, May 09 2015