cp's OEIS Frontend

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

%I A117995 #21 Aug 13 2025 23:45:00
%S A117995 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 A117995 383,477,574,707,847,1038,1238,1506,1794,2166,2573,3093,3660,4377,
%U A117995 5170,6152,7245,8590,10087,11913,13959,16423,19196,22518,26252,30700,35717
%N A117995 Number of partitions of n in which both smallest and largest part occur only once.
%C A117995 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 A117995 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 A117995 Seiichi Manyama, <a href="/A117995/b117995.txt">Table of n, a(n) for n = 1..10000</a>
%F A117995 G.f.: Sum_{k>=2} Sum_{j=1..k-1} x^(j+k)/Product_{i=j+1..k-1} (1-x^i).
%F A117995 G.f.: x^3/[(1-x)(1-x^2)] + Sum_{k>=3} x^(2k)/Product_{j=1..k} (1-x^j).
%F A117995 a(n+1) + A240078(n) = A240080(n) for n >= 0. - _Clark Kimberling_, Apr 01 2014
%F A117995 a(n) = A002865(n) - (n + 1) mod 2. - _Seiichi Manyama_, Jan 28 2022
%e A117995 a(8)=6 because we have [7,1],[6,2],[5,3],[5,2,1],[4,3,1] and [3,2,2,1].
%p A117995 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 A117995 (* See A240077. - _Clark Kimberling_, Apr 01 2014 *)
%t A117995 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 *)
%Y A117995 Cf. A002865, A240076.
%K A117995 nonn
%O A117995 1,5
%A A117995 _Emeric Deutsch_, Apr 08 2006