A257610 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 + 2.

1, 2, 2, 4, 20, 4, 8, 132, 132, 8, 16, 748, 2112, 748, 16, 32, 3964, 25124, 25124, 3964, 32, 64, 20364, 256488, 552728, 256488, 20364, 64, 128, 103100, 2398092, 9670840, 9670840, 2398092, 103100, 128, 256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256

Offset: 0

Author

Dale Gerdemann, May 03 2015

Keywords

• nonn
• tabl

Examples

			Triangle begins as:
    1;
    2,      2;
    4,     20,        4;
    8,    132,      132,         8;
   16,    748,     2112,       748,        16;
   32,   3964,    25124,     25124,      3964,        32;
   64,  20364,   256488,    552728,    256488,     20364,       64;
  128, 103100,  2398092,   9670840,   9670840,   2398092,   103100,    128;
  256, 518444, 21246736, 147146804, 270783520, 147146804, 21246736, 518444, 256;
		

Links

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

Crossrefs

Cf. A007559 (row sums), A038208, A142458, A257620, A257622, A257624, A257626.

Cf. A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257618, A257619.

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,3,2], {n,0,12}, {k,0,n}]//Flatten (* G. C. Greubel, Mar 20 2022 *)

Sage

def T(n,k,a,b): # A257610
    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,3,2) 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 + 2.

Sum_{k=0..n} T(n, k) = A007559(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 = 2. - G. C. Greubel, Mar 20 2022