A071417 Triangle of expected coupon collection numbers rounded up; i.e., if aiming to collect part of a set of n coupons, the expected number of random coupons required to receive first the set with exactly k missing.

0, 1, 0, 3, 1, 0, 6, 3, 1, 0, 9, 5, 3, 1, 0, 12, 7, 4, 3, 1, 0, 15, 9, 6, 4, 3, 1, 0, 19, 12, 8, 6, 4, 3, 1, 0, 22, 14, 10, 8, 6, 4, 3, 1, 0, 26, 17, 12, 9, 7, 5, 4, 3, 1, 0, 30, 20, 15, 11, 9, 7, 5, 4, 3, 1, 0, 34, 23, 17, 14, 11, 9, 7, 5, 4, 3, 1, 0, 38, 26, 20, 16, 13, 10, 8, 7, 5, 4, 3, 1, 0, 42

Offset: 0

Author

Henry Bottomley, May 29 2002

Keywords

• nonn
• tabl

Examples

			Rows start
0;
1,0;
3,1,0;
6,3,1,0;
9,5,3,1,0;
etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150).

Crossrefs

Cf. A060293 (left hand column), A067176.

Programs

Mathematica

Table[Ceiling[n Sum[1/j, {j, k + 1, n}]], {n, 0, 12}, {k, 0, n}] // Flatten (* Michael De Vlieger, Jul 30 2017 *)

Formula

a(n, k) = ceiling(n*Sum_{j=k+1..n} 1/j) = ceiling(A067176(n, k)*k!/(n-1)!) = ceiling(A008279(n, n-k)*Sum_{j>=n-k} j*A008277(j-1, n-k-1)/n^j).