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A116686 Total number of parts smaller than the largest part, in all partitions of n.

%I A116686 #23 Jan 23 2025 06:56:56
%S A116686 0,0,1,3,8,15,29,48,79,123,188,276,404,575,808,1122,1540,2089,2811,
%T A116686 3748,4958,6519,8504,11034,14231,18268,23312,29638,37486,47245,59279,
%U A116686 74140,92347,114703,141933,175174,215478,264407,323448,394788,480509,583609
%N A116686 Total number of parts smaller than the largest part, in all partitions of n.
%C A116686 Also, sum over all partitions of n of the difference between the largest part and the smallest part. - _Franklin T. Adams-Watters_, Feb 29 2008
%C A116686 Row sums of A244966. - _Omar E. Pol_, Jul 19 2014
%H A116686 Alois P. Heinz, <a href="/A116686/b116686.txt">Table of n, a(n) for n = 1..1000</a>
%F A116686 a(n) = Sum_{k>=0} k*A116685(n,k).
%F A116686 G.f.: Sum_{i>=1} (x^i*(Sum_{j=1..i-1} x^j/(1-x^j))/(Product_{q=1..i} (1-x^q))).
%F A116686 a(n) = A006128(n) - A046746(n). - _Vladeta Jovovic_, Feb 24 2006
%F A116686 a(n) = A211870(n) + A211881(n). - _Alois P. Heinz_, Feb 13 2013
%e A116686 a(5) = 8 because the partitions of 5 are [5], [4,(1)], [3,(2)], [3,(1),(1)], [2,2,(1)], [2,(1),(1),(1)] and [1,1,1,1,1], containing a total of 8 parts that are smaller than the largest part (shown between parentheses).
%p A116686 f:=sum(x^i*sum(x^j/(1-x^j),j=1..i-1)/product(1-x^q,q=1..i),i=2..55): fser:=series(f,x=0,50): seq(coeff(fser,x^n),n=1..47);
%t A116686 Table[Length[Flatten[Rest[Split[#]]&/@Select[IntegerPartitions[n], #[[1]]> #[[-1]]&]]],{n,50}] (* _Harvey P. Dale_, Jul 26 2016 *)
%Y A116686 Cf. A116685, A211870, A211881.
%K A116686 nonn
%O A116686 1,4
%A A116686 _Emeric Deutsch_, Feb 23 2006