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).

1, 1, 2, 3, 4, 6, 13, 16, 20, 26, 75, 88, 104, 124, 150, 541, 616, 704, 808, 932, 1082, 4683, 5224, 5840, 6544, 7352, 8284, 9366, 47293, 51976, 57200, 63040, 69584, 76936, 85220, 94586, 545835, 593128, 645104, 702304, 765344, 834928, 911864

Offset: 0

Author

Paul D. Hanna, Jul 18 2002

Keywords

• easy
• nonn
• tabl

Comments

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

Row sums are 1, 3, 13, 75, 541, ... (A000670 starting from A000670(1), the second "1"). - Gary W. Adamson, May 31 2005

Examples

			Triangle begins:
    1;
    1,   2;
    3,   4,   6;
   13,  16,  20,  26;
   75,  88, 104, 124, 150;
  541, 616, 704, 808, 932, 1082;
  ...
		

Links

• D. Dumont, Matrices d'Euler-Seidel, Sem. Loth. Comb. B05c (1981) 59-78.

Crossrefs

Cf. A000670, A000629, A011971.

Main diagonal is in A098696.

Programs

Mathematica

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;
a[n_, k_] := Sum[Binomial[k, i-n+k] Fubini[i, 1], {i, n-k, n}];
Table[a[n, k], {n, 0, 8}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 30 2016, after Vladeta Jovovic *)

Formula

From Vladeta Jovovic, Oct 15 2006: (Start)

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

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