A257618 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) = 8*x + 2.

1, 2, 2, 4, 40, 4, 8, 472, 472, 8, 16, 4928, 16992, 4928, 16, 32, 49824, 433984, 433984, 49824, 32, 64, 499584, 9505728, 22567168, 9505728, 499584, 64, 128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128

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) = 8*x + 2.

Sum_{k=0..n} T(n, k) = A144828(n).

From G. C. Greubel, Mar 24 2022: (Start)

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 = 8, and b = 2.

T(n, n-k) = T(n, k).

T(n, 0) = A000079(n).

T(n, 1) = 2^(n-1)*(5^n - 2*n - 1).

T(n, 2) = 2^(n-3)*(3^(2*n+1) -2*(2*n+1)*5^n -1 +4*n^2). (End)

Example

Triangle begins as:
    1;
    2,       2;
    4,      40,         4;
    8,     472,       472,         8;
   16,    4928,     16992,      4928,        16;
   32,   49824,    433984,    433984,     49824,        32;
   64,  499584,   9505728,  22567168,   9505728,    499584,      64;
  128, 4999040, 192085632, 909941120, 909941120, 192085632, 4999040, 128;

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, 8, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

Prog

(Sage)
def T(n, k, a, b): # A257618
    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, 8, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

Crossrefs

Cf. A000079, A144828 (row sums), A167884.

Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257617, A257619.

Similar sequences listed in A256890.

Sequence in context: A300361 A257617 A309344 * A088895 A257619 A075806

Adjacent sequences: A257615 A257616 A257617 * A257619 A257620 A257621

Keyword

nonn,tabl

Author

Dale Gerdemann, May 09 2015

Status

approved