A073146 - OEIS

%I #19 Feb 06 2022 21:18:07

%S 1,1,2,3,4,6,13,16,20,26,75,88,104,124,150,541,616,704,808,932,1082,

%T 4683,5224,5840,6544,7352,8284,9366,47293,51976,57200,63040,69584,

%U 76936,85220,94586,545835,593128,645104,702304,765344,834928,911864

%N 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 Related to preferential arrangements of n elements (A000670) and necklaces of sets of labeled beads (A000629).

%C Row sums are 1, 3, 13, 75, 541, ... (A000670 starting from A000670(1), the second "1"). - _Gary W. Adamson_, May 31 2005

%H 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 From _Vladeta Jovovic_, Oct 15 2006: (Start)

%F Double-exponential generating function: Sum_{n, k} a(n-k, k) x^n/n! y^k/k! = exp(y)/(2-exp(x+y)).

%F a(n,k) = Sum_{i=n-k..n} binomial(k,i-n+k)*A000670(i). (End)

%e Triangle begins:

%e     1;

%e     1,   2;

%e     3,   4,   6;

%e    13,  16,  20,  26;

%e    75,  88, 104, 124, 150;

%e   541, 616, 704, 808, 932, 1082;

%e   ...

%t 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 a[n_, k_] := Sum[Binomial[k, i-n+k] Fubini[i, 1], {i, n-k, n}];

%t Table[a[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Mar 30 2016, after _Vladeta Jovovic_ *)

%Y Cf. A000670, A000629, A011971.

%Y Main diagonal is in A098696.

%K easy,nonn,tabl

%O 0,3

%A _Paul D. Hanna_, Jul 18 2002