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
Examples
1; 0, 1; 1, 1, 2; 1, 2, 3, 5; 4, 5, 7, 10, 15; 11, 15, 20, 27, 37, 52; ...
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.
Crossrefs
Programs
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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 *) a[0,0]:=1; a[n_,0]:=a[n-1,n-1]-a[n-1,0]; a[n_,k_]/;0
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
Comments