A257622 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 + 4.

1, 4, 4, 16, 56, 16, 64, 552, 552, 64, 256, 4696, 11040, 4696, 256, 1024, 36968, 171448, 171448, 36968, 1024, 4096, 278232, 2305968, 4457648, 2305968, 278232, 4096, 16384, 2037736, 28346088, 94844912, 94844912, 28346088, 2037736, 16384

Offset

0,2

Links

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

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 + 4.

Sum_{k=0..n} T(n, k) = A051605(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 = 4. - G. C. Greubel, Mar 20 2022

Example

Triangle begins as:
      1;
      4,       4;
     16,      56,       16;
     64,     552,      552,       64;
    256,    4696,    11040,     4696,      256;
   1024,   36968,   171448,   171448,    36968,     1024;
   4096,  278232,  2305968,  4457648,  2305968,   278232,    4096;
  16384, 2037736, 28346088, 94844912, 94844912, 28346088, 2037736, 16384;

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

Prog

(Sage)
def T(n, k, a, b): # A257622
    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, 4) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 20 2022

Crossrefs

Cf. A051605 (row sums), A142458, A257610, A257620, A257624, A257626.

Cf. A257606, A257613.

See similar sequences listed in A256890.

Sequence in context: A222144 A342817 A107382 * A185567 A203536 A038788

Adjacent sequences: A257619 A257620 A257621 * A257623 A257624 A257625

Keyword

nonn,tabl

Author

Dale Gerdemann, May 10 2015

Status

approved