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A257615 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) = 2*x + 5.

%I A257615 #14 Mar 22 2022 03:11:31
%S A257615 1,5,5,25,70,25,125,715,715,125,625,6380,12870,6380,625,3125,52785,
%T A257615 186010,186010,52785,3125,15625,416370,2360295,4092220,2360295,416370,
%U A257615 15625,78125,3180215,27488205,75698255,75698255,27488205,3180215,78125
%N A257615 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) = 2*x + 5.
%H A257615 G. C. Greubel, <a href="/A257615/b257615.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257615 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) = 2*x + 5.
%F A257615 Sum_{k=0..n} T(n, k) = A051582(n).
%F A257615 From _G. C. Greubel_, Mar 21 2022: (Start)
%F A257615 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 = 2, and b = 5.
%F A257615 T(n, n-k) = T(n, k).
%F A257615 T(n, 0) = A000351(n).
%F A257615 T(n, 1) = 5*7^n - 5^n*(n+5). (End)
%e A257615 Triangle begins as:
%e A257615       1;
%e A257615       5,       5;
%e A257615      25,      70,       25;
%e A257615     125,     715,      715,      125;
%e A257615     625,    6380,    12870,     6380,      625;
%e A257615    3125,   52785,   186010,   186010,    52785,     3125;
%e A257615   15625,  416370,  2360295,  4092220,  2360295,   416370,   15625;
%e A257615   78125, 3180215, 27488205, 75698255, 75698255, 27488205, 3180215, 78125;
%t A257615 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 A257615 Table[T[n,k,2,5], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 21 2022 *)
%o A257615 (Sage)
%o A257615 def T(n,k,a,b): # A257610
%o A257615     if (k<0 or k>n): return 0
%o A257615     elif (n==0): return 1
%o A257615     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257615 flatten([[T(n,k,2,5) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 21 2022
%Y A257615 Cf. A000351, A051582 (row sums), A060187, A257609, A257611, A257613.
%Y A257615 Cf. A257607, A257624.
%Y A257615 Similar sequences listed in A256890.
%K A257615 nonn,tabl
%O A257615 0,2
%A A257615 _Dale Gerdemann_, May 09 2015