A106436 Difference array of Bell numbers A000110 read by antidiagonals.

1, 0, 1, 1, 1, 2, 1, 2, 3, 5, 4, 5, 7, 10, 15, 11, 15, 20, 27, 37, 52, 41, 52, 67, 87, 114, 151, 203, 162, 203, 255, 322, 409, 523, 674, 877, 715, 877, 1080, 1335, 1657, 2066, 2589, 3263, 4140, 3425, 4140, 5017, 6097, 7432, 9089, 11155, 13744, 17007, 21147

Offset

0,6

Comments

Essentially Aitken's array A011971 with first column A000296.

Mirror image of A182930. - Alois P. Heinz, Jan 29 2019

Links

Alois P. Heinz, Rows n = 0..140, flattened

Jocelyn Quaintance and Harris Kwong, A combinatorial interpretation of the Catalan and Bell number difference tables, Integers, 13 (2013), #A29.

Formula

Double-exponential generating function: sum_{n, k} a(n-k, k) x^n/n! y^k/k! = exp(exp{x+y}-1-x). a(n,k) = Sum_{i=k..n} (-1)^(n-i)*binomial(n-k,i-k)*Bell(i). - Vladeta Jovovic, Oct 14 2006

Example

   1;
   0,  1;
   1,  1,  2;
   1,  2,  3,  5;
   4,  5,  7, 10, 15;
  11, 15, 20, 27, 37, 52;
  ...

Maple

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-j)*binomial(n-1, j-1), j=1..n))
    end:
T:= proc(n, k) option remember; `if`(k=0, b(n),
      T(n+1, k-1)-T(n, k-1))
    end:
seq(seq(T(n, d-n), n=0..d), d=0..12);  # Alois P. Heinz, Jan 29 2019

Mathematica

bb = Array[BellB, m = 12, 0];
dd[n_] := Differences[bb, n];
A = Array[dd, m, 0];
Table[A[[n-k+1, k+1]], {n, 0, m-1}, {k, 0, n}] // Flatten (* Jean-François Alcover, May 26 2019 *)

Crossrefs

Cf. A000110, A182930.

Diagonals give A005493, A011965-A011967, A191099, A000298, A011968-A011970.

T(2n,n) gives A020556.

Sequence in context: A274491 A076492 A127462 * A363226 A075758 A125596

Adjacent sequences: A106433 A106434 A106435 * A106437 A106438 A106439

Keyword

nonn,easy,tabl

Author

Philippe Deléham, May 29 2005

Status

approved