A118084 Number of partitions of n such that largest part k occurs at most floor(k/2) times.

0, 1, 2, 3, 5, 7, 11, 16, 23, 33, 46, 63, 86, 116, 153, 203, 265, 345, 444, 571, 727, 925, 1166, 1468, 1836, 2293, 2845, 3525, 4345, 5347, 6550, 8011, 9758, 11867, 14380, 17399, 20984, 25269, 30341, 36376, 43500, 51943, 61877, 73608, 87373, 103571

Offset: 1

Author

Emeric Deutsch, Apr 12 2006

Keywords

• nonn

Comments

Also number of partitions of n such that if the number of parts is k, then the smallest part is at most floor(k/2). Example: a(8)=16 because we have [7,1],[6,1,1],[5,2,1],[4,3,1],[5,1,1,1],[4,2,1,1],[3,3,1,1],[3,2,2,1],[2,2,2,2],[4,1,1,1,1],[3,2,1,1,1],[2,2,2,1,1],[3,1,1,1,1,1],[2,2,1,1,1,1],[2,1,1,1,1,1,1] and [1,1,1,1,1,1,1,1].

Examples

			a(8)=16 because we have [8],[7,1],[6,2],[6,1,1],[5,3],[5,2,1],[5,1,1,1],[4,4],[4,3,1],[4,2,2],[4,2,1,1],[4,1,1,1,1],[3,2,2,1],[3,2,1,1,1],[3,1,1,1,1,1] and [2,1,1,1,1,1,1].
		

Crossrefs

Cf. A118082, A118083.

Cf. A237758, A237757, A237799, A237800. - Clark Kimberling, Feb 15 2014

Programs

Maple

g:=sum(x^k*(1-x^(k*(floor(k/2))))/product(1-x^j,j=1..k),k=1..85): gser:=series(g,x=0,55): seq(coeff(gser,x,n),n=1..50);

Mathematica

z=55 ; q[n_] := q[n] = IntegerPartitions[n]; t[p_] := Length[p];
Table[Count[q[n], p_ /; 2 Min[p] <= t[p]], {n,z}] (* Clark Kimberling, Feb 15 2014 *)

Formula

G.f.=sum(x^k*(1-x^(k(floor(k/2))))/product(1-x^j, j=1..k), k=1..infinity).