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A257624 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) = 3*x + 5.

%I A257624 #14 Mar 21 2022 02:14:27
%S A257624 1,5,5,25,80,25,125,915,915,125,625,9070,20130,9070,625,3125,83185,
%T A257624 348410,348410,83185,3125,15625,727980,5246655,9755480,5246655,727980,
%U A257624 15625,78125,6183215,72272805,225769855,225769855,72272805,6183215,78125
%N A257624 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) = 3*x + 5.
%H A257624 G. C. Greubel, <a href="/A257624/b257624.txt">Rows n = 0..50 of the triangle, flattened</a>
%F A257624 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) = 3*x + 5.
%F A257624 Sum_{k=0..n} T(n, k) = A051607(n).
%F A257624 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 = 3, and b = 5. - _G. C. Greubel_, Mar 20 2022
%e A257624 Triangle begins as:
%e A257624       1;
%e A257624       5,       5;
%e A257624      25,      80,       25;
%e A257624     125,     915,      915,       125;
%e A257624     625,    9070,    20130,      9070,       625;
%e A257624    3125,   83185,   348410,    348410,     83185,     3125;
%e A257624   15625,  727980,  5246655,   9755480,   5246655,   727980,   15625;
%e A257624   78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 78125;
%t A257624 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 A257624 Table[T[n,k,3,5], {n,0,12}, {k,0,n}]//Flatten (* _G. C. Greubel_, Mar 20 2022 *)
%o A257624 (Sage)
%o A257624 def T(n,k,a,b): # A257624
%o A257624     if (k<0 or k>n): return 0
%o A257624     elif (n==0): return 1
%o A257624     else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
%o A257624 flatten([[T(n,k,3,5) for k in (0..n)] for n in (0..12)]) # _G. C. Greubel_, Mar 20 2022
%Y A257624 Cf. A051607 (row sums), A142458, A257610, A257620, A257622, A257626.
%Y A257624 Cf. A257607, A257615.
%Y A257624 Similar sequences listed in A256890.
%K A257624 nonn,tabl
%O A257624 0,2
%A A257624 _Dale Gerdemann_, May 10 2015