A117995 Number of partitions of n in which both smallest and largest part occur only once.

0, 0, 1, 1, 2, 3, 4, 6, 8, 11, 14, 20, 24, 33, 41, 54, 66, 87, 105, 136, 165, 209, 253, 319, 383, 477, 574, 707, 847, 1038, 1238, 1506, 1794, 2166, 2573, 3093, 3660, 4377, 5170, 6152, 7245, 8590, 10087, 11913, 13959, 16423, 19196, 22518, 26252, 30700, 35717

Offset: 1

Author

Emeric Deutsch, Apr 08 2006

Keywords

• nonn

Comments

Also number of partitions of n in which the least part is 1 and if k is the largest part, then k>=2 and k-1 also occurs. Example: a(8)=6 because we have [4,3,1],[3,2,2,1],[3,2,1,1,1],[2,2,2,1,1],[2,2,1,1,1,1] and [2,1,1,1,1,1,1].

a(n+1) is the number of partitions of n such that m(greatest part) > m(1), where m = multiplicity, for n>= 0. For example, a(8) counts these 6 partitions of 7: 7, 52, 43, 331, 322, 2221. - Clark Kimberling, Apr 01 2014

Examples

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

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002865, A240076.

Programs

Maple

g:=x^3/(1-x)/(1-x^2)+sum(x^(2*k)/product(1-x^j,j=1..k),k=3..70): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=1..55);

Mathematica

(* See A240077. - Clark Kimberling, Apr 01 2014 *)
sl1Q[n_]:=With[{c=Split[n]},Length[c]>1&&Length[c[[1]]]==Length[c[[-1]]==1]]; Table[Count[IntegerPartitions[n],?(sl1Q)],{n,3,60}] (* _Harvey P. Dale, Apr 06 2025 *)

Formula

G.f.: Sum_{k>=2} Sum_{j=1..k-1} x^(j+k)/Product_{i=j+1..k-1} (1-x^i).

G.f.: x^3/[(1-x)(1-x^2)] + Sum_{k>=3} x^(2k)/Product_{j=1..k} (1-x^j).

a(n+1) + A240078(n) = A240080(n) for n >= 0. - Clark Kimberling, Apr 01 2014

a(n) = A002865(n) - (n + 1) mod 2. - Seiichi Manyama, Jan 28 2022