A257626 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 + 6.

1, 6, 6, 36, 108, 36, 216, 1404, 1404, 216, 1296, 15876, 33696, 15876, 1296, 7776, 166212, 642492, 642492, 166212, 7776, 46656, 1659204, 10701720, 19274760, 10701720, 1659204, 46656, 279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936

Offset: 0

Author

Dale Gerdemann, May 10 2015

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
       1;
       6,        6;
      36,      108,        36;
     216,     1404,      1404,       216;
    1296,    15876,     33696,     15876,      1296;
    7776,   166212,    642492,    642492,    166212,      7776;
   46656,  1659204,  10701720,  19274760,  10701720,   1659204,    46656;
  279936, 16052580, 163263924, 481752360, 481752360, 163263924, 16052580, 279936;
		

Links

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

Crossrefs

Cf. A051609 (row sums), A142458, A257610, A257620, A257622, A257624.

See 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,3,6], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

Sage

def T(n,k,a,b): # A257626
    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,3,6) 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 + 6.

Sum_{k=0..n} T(n, k) = A051609(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 = 6. - G. C. Greubel, Mar 20 2022