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