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

0,4

Links

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

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).

Example

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;

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 *)

Prog

(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
(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

Crossrefs

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

Rows include: A000012, A117142.

Sequence in context: A271025 A134379 A108087 * A185414 A346520 A362925

Adjacent sequences: A123155 A123156 A123157 * A123159 A123160 A123161

Keyword

easy,nonn,tabl

Author

Philippe Deléham, Oct 01 2006

Status

approved