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

1,5

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

Links

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

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

Example

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

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

Crossrefs

Cf. A002865, A240076.

Sequence in context: A094707 A353864 A303663 * A033834 A127419 A262160

Adjacent sequences: A117992 A117993 A117994 * A117996 A117997 A117998

Keyword

nonn

Author

Emeric Deutsch, Apr 08 2006

Status

approved