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

Author

Dale Gerdemann, May 03 2015

Keywords

• nonn
• tabl

Examples

			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;
		

Links

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

Crossrefs

Cf. A000079, A002866 (row sums), A008292, A060187, A100575, A257611, A257613.

Cf. A257615, A038208, A256890, A257610, A257612, A257614, A257616, A257617.

Cf. A257618, A257619.

Similar sequences listed in A256890.

Programs

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

Magma

A257609:= func< n,k | 2^n*EulerianNumber(n+1, k) >;
[A257609(n,k): k in [0..n], n in [0..10]]; // G. C. Greubel, Jan 17 2025

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 *)

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

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)

T(n, k) = 2^n*A008292(n+1, k+1). - G. C. Greubel, Jan 17 2025