A123158 Square array related to Bell numbers read by antidiagonals.

1, 1, 1, 2, 2, 1, 5, 5, 3, 1, 15, 15, 10, 5, 1, 52, 52, 37, 22, 6, 1, 203, 203, 151, 99, 31, 9, 1, 877, 877, 674, 471, 160, 61, 10, 1, 4140, 4140, 3263, 2386, 856, 385, 75, 14, 1, 21147, 21147, 17007, 12867, 4802, 2416, 520, 135, 15, 1

Offset: 0

Author

Philippe Deléham, Oct 01 2006

Keywords

• easy
• nonn
• tabl

Examples

			Square array, A(n, k), begins:
   1,   1,   1,    1,    1, ... (Row n=0: A000012);
   1,   2,   3,    5,    6, ... (Row n=1: A117142);
   2,   5,  10,   22,   31, ...;
   5,  15,  37,   99,  160, ...;
  15,  52, 151,  471,  856, ...;
  52, 203, 674, 2386, 4802, ...;
Antidiagonals, T(n, k), begin as:
    1;
    1,   1;
    2,   2,   1;
    5,   5,   3,   1;
   15,  15,  10,   5,   1;
   52,  52,  37,  22,   6,  1;
  203, 203, 151,  99,  31,  9,   1;
  877, 877, 674, 471, 160, 61,  10,  1;
		

Links

• G. C. Greubel, Antidiagonals n = 0..50, flattened

Crossrefs

Columns include: A000110 (Bell numbers), A003128, A005493, A033452.

Rows include: A000012, A117142.

Programs

Magma

function A(n,k)
  if k lt 0 or n lt 0 then return 0;
  elif n eq 0 then return 1;
  elif (k mod 2) eq 1 then return A(n,k-1) + (1/2)*(k+1)*A(n-1,k+1);
  else return A(n,k-1) + A(n-1,k+1);
  end if;
end function;
T:= func< n,k | A(n-k,k) >;
[T(n,k): k in [0..n], n in [0..12]]; // G. C. Greubel, Jul 18 2023

Mathematica

A[0, _?NonNegative] = 1;
A[n_, k_]:= A[n, k]= If[n<0 || k<0, 0, If[OddQ[k], A[n, k-1] + (1/2)(k+1) A[n-1, k+1], A[n, k-1] + A[n-1, k+1]]];
Table[A[n-k, k], {n,0,10}, {k,0,n}]//Flatten (* Jean-François Alcover, Feb 21 2020 *)

SageMath

@CachedFunction
def A(n,k):
    if (k<0 or n<0): return 0
    elif (n==0): return 1
    elif (k%2==1): return A(n,k-1) +(1/2)*(k+1)*A(n-1,k+1)
    else: return A(n,k-1) +A(n-1,k+1)
def T(n,k): return A(n-k,k)
flatten([[T(n,k) for k in range(n+1)] for n in range(12)]) # G. C. Greubel, Jul 18 2023

Formula

A(n, k) = 0 if n < 0, A(0, k) = 1 for k >= 0, A(n, k) = A(n, k-1) + (1/2)*(k+1)*A(n-1, k+1) if k is an odd number, A(n, k) = A(n, k-1) + A(n-1, k+1) if k is an even number (array).

A(n, 0) = A000110(n).

A(n, 1) = A000110(n+1).

A(n, 2) = A005493(n).

A(n, 3) = A033452(n).

A(n, 4) = A003128(n+2).

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