A257617 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) = 7*x + 2.

1, 2, 2, 4, 36, 4, 8, 388, 388, 8, 16, 3676, 12416, 3676, 16, 32, 33564, 283204, 283204, 33564, 32, 64, 303260, 5538184, 13027384, 5538184, 303260, 64, 128, 2732156, 99831564, 465775352, 465775352, 99831564, 2732156, 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) = 7*x + 2.

Sum_{k=0..n} T(n, k) = A144827(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 = 7, and b = 2.

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

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

T(n, 1) = (4*9^n - 2^n*(7*n + 4))/7.

T(n, 2) = (2^(n-1)*(49*n^2 +7*n -12) + 11*2^(4*n+1) - 4*(7*n+4)*9^n)/49. (End)

Example

    1;
    2,       2;
    4,      36,        4;
    8,     388,      388,         8;
   16,    3676,    12416,      3676,        16;
   32,   33564,   283204,    283204,     33564,       32;
   64,  303260,  5538184,  13027384,   5538184,   303260,      64;
  128, 2732156, 99831564, 465775352, 465775352, 99831564, 2732156, 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, 7, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 24 2022 *)

Prog

(Sage)
def T(n, k, a, b): # A257617
    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, 7, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 24 2022

Crossrefs

Cf. A000079, A142462, A144827 (row sums), A257627.

Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257616, A257618, A257619

Similar sequences listed in A256890.

Sequence in context: A296048 A327011 A300361 * A309344 A257618 A088895

Adjacent sequences: A257614 A257615 A257616 * A257618 A257619 A257620

Keyword

nonn,tabl

Author

Dale Gerdemann, May 09 2015

Status

approved