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A106436 Difference array of Bell numbers A000110 read by antidiagonals.

%I A106436 #34 Nov 23 2024 09:04:15
%S A106436 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,
%T A106436 151,203,162,203,255,322,409,523,674,877,715,877,1080,1335,1657,2066,
%U A106436 2589,3263,4140,3425,4140,5017,6097,7432,9089,11155,13744,17007,21147
%N A106436 Difference array of Bell numbers A000110 read by antidiagonals.
%C A106436 Essentially Aitken's array A011971 with first column A000296.
%C A106436 Mirror image of A182930. - _Alois P. Heinz_, Jan 29 2019
%H A106436 Alois P. Heinz, <a href="/A106436/b106436.txt">Rows n = 0..140, flattened</a>
%H A106436 Jocelyn Quaintance and Harris Kwong, <a href="http://www.emis.de/journals/INTEGERS/papers/n29/n29.Abstract.html">A combinatorial interpretation of the Catalan and Bell number difference tables</a>, Integers, 13 (2013), #A29.
%F A106436 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
%e A106436    1;
%e A106436    0,  1;
%e A106436    1,  1,  2;
%e A106436    1,  2,  3,  5;
%e A106436    4,  5,  7, 10, 15;
%e A106436   11, 15, 20, 27, 37, 52;
%e A106436   ...
%p A106436 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A106436       b(n-j)*binomial(n-1, j-1), j=1..n))
%p A106436     end:
%p A106436 T:= proc(n, k) option remember; `if`(k=0, b(n),
%p A106436       T(n+1, k-1)-T(n, k-1))
%p A106436     end:
%p A106436 seq(seq(T(n, d-n), n=0..d), d=0..12);  # _Alois P. Heinz_, Jan 29 2019
%t A106436 bb = Array[BellB, m = 12, 0];
%t A106436 dd[n_] := Differences[bb, n];
%t A106436 A = Array[dd, m, 0];
%t A106436 Table[A[[n-k+1, k+1]], {n, 0, m-1}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, May 26 2019 *)
%t A106436 a[0,0]:=1; a[n_,0]:=a[n-1,n-1]-a[n-1,0]; a[n_,k_]/;0<k<=n:=a[n,k]=a[n,k-1]+a[n-1,k-1];
%t A106436 a[n_,k_]:=0; Flatten@Table[a[n,k],{n,0,9},{k,0,n}] (* _Oliver Seipel_, Nov 23 2024 *)
%Y A106436 Cf. A000110, A182930.
%Y A106436 Diagonals give A005493, A011965-A011967, A191099, A000298, A011968-A011970.
%Y A106436 T(2n,n) gives A020556.
%K A106436 nonn,easy,tabl
%O A106436 0,6
%A A106436 _Philippe Deléham_, May 29 2005