A117995 - OEIS

%I #15 Jan 28 2022 16:43:26

%S 0,0,1,1,2,3,4,6,8,11,14,20,24,33,41,54,66,87,105,136,165,209,253,319,

%T 383,477,574,707,847,1038,1238,1506,1794,2166,2573,3093,3660,4377,

%U 5170,6152,7245,8590,10087,11913,13959,16423,19196,22518,26252,30700,35717

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

%C 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].

%C 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

%H Seiichi Manyama, <a href="/A117995/b117995.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F a(n+1) + A240078(n) = A240080(n) for n >= 0. - _Clark Kimberling_, Apr 01 2014

%F a(n) = A002865(n) - (n + 1) mod 2. - _Seiichi Manyama_, Jan 28 2022

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

%p 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);

%t (See A240077.)  - _Clark Kimberling_, Apr 01 2014

%Y Cf. A002865, A240076.

%K nonn

%O 1,5

%A _Emeric Deutsch_, Apr 08 2006