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.

1, 5, 5, 25, 80, 25, 125, 915, 915, 125, 625, 9070, 20130, 9070, 625, 3125, 83185, 348410, 348410, 83185, 3125, 15625, 727980, 5246655, 9755480, 5246655, 727980, 15625, 78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 78125

Offset: 0

Author

Dale Gerdemann, May 10 2015

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
      1;
      5,       5;
     25,      80,       25;
    125,     915,      915,       125;
    625,    9070,    20130,      9070,       625;
   3125,   83185,   348410,    348410,     83185,     3125;
  15625,  727980,  5246655,   9755480,   5246655,   727980,   15625;
  78125, 6183215, 72272805, 225769855, 225769855, 72272805, 6183215, 78125;
		

Links

• G. C. Greubel, Rows n = 0..50 of the triangle, flattened

Crossrefs

Cf. A051607 (row sums), A142458, A257610, A257620, A257622, A257626.

Cf. A257607, A257615.

Similar sequences listed in A256890.

Programs

Mathematica

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]]];
Table[T[n,k,3,5], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

Sage

def T(n,k,a,b): # A257624
    if (k<0 or k>n): return 0
    elif (n==0): return 1
    else: return  (a*k+b)*T(n-1,k,a,b) + (a*(n-k)+b)*T(n-1,k-1,a,b)
flatten([[T(n,k,3,5) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Formula

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.

Sum_{k=0..n} T(n, k) = A051607(n).

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