A257609 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) = 2*x + 2.

1, 2, 2, 4, 16, 4, 8, 88, 88, 8, 16, 416, 1056, 416, 16, 32, 1824, 9664, 9664, 1824, 32, 64, 7680, 76224, 154624, 76224, 7680, 64, 128, 31616, 549504, 1999232, 1999232, 549504, 31616, 128, 256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 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) = 2*x + 2.

Sum_{k=0..n} T(n, k) = A002866(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 = 2, and b = 2.

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

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

T(n, 1) = 2*A100575(n+1). (End)

Example

Triangle begins as:
    1;
    2,      2;
    4,     16,       4;
    8,     88,      88,        8;
   16,    416,    1056,      416,       16;
   32,   1824,    9664,     9664,     1824,       32;
   64,   7680,   76224,   154624,    76224,     7680,      64;
  128,  31616,  549504,  1999232,  1999232,   549504,   31616,    128;
  256, 128512, 3739648, 22587904, 39984640, 22587904, 3739648, 128512, 256;

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, 2, 2], {n, 0, 12}, {k, 0, n}]//Flatten (* G. C. Greubel, Mar 21 2022 *)

Prog

(Magma)
function T(n, k, a, b)
  if k lt 0 or k gt n then return 0;
  elif k eq 0 or k eq n then return 1;
  else return (a*k+b)*T(n-1, k, a, b) + (a*(n-k)+b)*T(n-1, k-1, a, b);
  end if; return T;
end function;
[T(n, k, 2, 2): k in [0..n], n in [0..12]]; // G. C. Greubel, Mar 21 2022
(Sage)
def T(n, k, a, b): # A257609
    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, 2, 2) for k in (0..n)] for n in (0..12)]) # G. C. Greubel, Mar 21 2022

Crossrefs

Cf. A000079, A002866 (row sums), A060187, A100575, A257611, A257613, A257615.

Cf. A038208, A256890, A257610, A257612, A257614, A257616, A257617, A257618, A257619.

Cf. similar sequences listed in A256890.

Sequence in context: A121221 A122598 A279069 * A087783 A176190 A106241

Adjacent sequences: A257606 A257607 A257608 * A257610 A257611 A257612

Keyword

nonn,tabl

Author

Dale Gerdemann, May 03 2015

Status

approved