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A073146 Triangle of numbers {a(n,k), n >= 0, 0 <= k <= n} defined by a(0,0)=1, a(n,0)=A000670(n), a(n,n)=A000629(n), a(n,k) = a(n,k-1) + a(n-1,k-1); a(n+1,0) = Sum_{k=0..n} a(n,k).

%I A073146 #19 Feb 06 2022 21:18:07
%S A073146 1,1,2,3,4,6,13,16,20,26,75,88,104,124,150,541,616,704,808,932,1082,
%T A073146 4683,5224,5840,6544,7352,8284,9366,47293,51976,57200,63040,69584,
%U A073146 76936,85220,94586,545835,593128,645104,702304,765344,834928,911864
%N A073146 Triangle of numbers {a(n,k), n >= 0, 0 <= k <= n} defined by a(0,0)=1, a(n,0)=A000670(n), a(n,n)=A000629(n), a(n,k) = a(n,k-1) + a(n-1,k-1); a(n+1,0) = Sum_{k=0..n} a(n,k).
%C A073146 Related to preferential arrangements of n elements (A000670) and necklaces of sets of labeled beads (A000629).
%C A073146 Row sums are 1, 3, 13, 75, 541, ... (A000670 starting from A000670(1), the second "1"). - _Gary W. Adamson_, May 31 2005
%H A073146 D. Dumont, <a href="http://www.mat.univie.ac.at/~slc/opapers/s05dumont.html">Matrices d'Euler-Seidel</a>, Sem. Loth. Comb. B05c (1981) 59-78.
%F A073146 From _Vladeta Jovovic_, Oct 15 2006: (Start)
%F A073146 Double-exponential generating function: Sum_{n, k} a(n-k, k) x^n/n! y^k/k! = exp(y)/(2-exp(x+y)).
%F A073146 a(n,k) = Sum_{i=n-k..n} binomial(k,i-n+k)*A000670(i). (End)
%e A073146 Triangle begins:
%e A073146     1;
%e A073146     1,   2;
%e A073146     3,   4,   6;
%e A073146    13,  16,  20,  26;
%e A073146    75,  88, 104, 124, 150;
%e A073146   541, 616, 704, 808, 932, 1082;
%e A073146   ...
%t A073146 Fubini[n_, r_] := Sum[k!*Sum[(-1)^(i+k+r)*(i+r)^(n-r)/(i!*(k-i-r)!), {i, 0, k-r}], {k, r, n}]; Fubini[0, 1] = 1;
%t A073146 a[n_, k_] := Sum[Binomial[k, i-n+k] Fubini[i, 1], {i, n-k, n}];
%t A073146 Table[a[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 30 2016, after _Vladeta Jovovic_ *)
%Y A073146 Cf. A000670, A000629, A011971.
%Y A073146 Main diagonal is in A098696.
%K A073146 easy,nonn,tabl
%O A073146 0,3
%A A073146 _Paul D. Hanna_, Jul 18 2002