A257616 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) = 6*x + 2.

1, 2, 2, 4, 32, 4, 8, 312, 312, 8, 16, 2656, 8736, 2656, 16, 32, 21664, 175424, 175424, 21664, 32, 64, 174336, 3019200, 7016960, 3019200, 174336, 64, 128, 1397120, 47847552, 218838400, 218838400, 47847552, 1397120, 128, 256, 11182592, 722956288, 5907889664, 11379596800, 5907889664, 722956288, 11182592, 256

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) = 6*x + 2.

Sum_{k=0..n} T(n, k) = A049308(n).

From G. C. Greubel, Mar 21 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 = 6, and b = 2.

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

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

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

Example

Triangle begins as:
    1;
    2,       2;
    4,      32,        4;
    8,     312,      312,         8;
   16,    2656,     8736,      2656,        16;
   32,   21664,   175424,    175424,     21664,       32;
   64,  174336,  3019200,   7016960,   3019200,   174336,      64;
  128, 1397120, 47847552, 218838400, 218838400, 47847552, 1397120, 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, 6, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

Prog

(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, 6, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

Crossrefs

Cf. A000079, A049308 (row sums), A142461, A257625.

Cf. A038208, A256890, A257609, A257610, A257612, A257614, A257617, A257618, A257619.

Similar sequences listed in A256890.

Sequence in context: A266046 A032334 A032082 * A296048 A327011 A300361

Adjacent sequences: A257613 A257614 A257615 * A257617 A257618 A257619

Keyword

nonn,tabl

Author

Dale Gerdemann, May 09 2015

Status

approved