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A116688 Sum over all partitions of n of the sum of the parts that are smaller than the largest part.

%I A116688 #8 Jan 23 2025 06:49:26
%S A116688 0,0,1,3,9,17,36,61,106,171,273,411,627,916,1326,1890,2667,3698,5102,
%T A116688 6943,9388,12588,16747,22113,29051,37914,49191,63515,81589,104315,
%U A116688 132799,168351,212540,267395,335085,418574,521093,646763,800164,987315
%N A116688 Sum over all partitions of n of the sum of the parts that are smaller than the largest part.
%F A116688 G.f.=sum(x^i*sum(jx^j/(1-x^j), j=1..i-1)/product(1-x^q, q=1..i), i=1..infinity).
%F A116688 a(n) = Sum_{k>=0} k * A116687(n,k).
%F A116688 a(n) = n*A000041(n) - A092321(n). - _Vladeta Jovovic_, Feb 24 2006
%e A116688 a(5)=9 because the partitions of 5 are [5],[4,(1)],[3,(2)],[3,(1),(1)],
%e A116688 [2,2,(1)],[2,(1),(1),(1)] and [1,1,1,1,1] and the sum of the parts (shown between parentheses) that are smaller than the largest part is 1+2+1+1+1+1+1+1=9.
%p A116688 f:=sum(x^i*sum(j*x^j/(1-x^j),j=1..i-1)/product(1-x^q,q=1..i),i=1..55): fser:=series(f,x=0,50): seq(coeff(fser,x^n),n=1..47);
%Y A116688 Cf. A116687.
%K A116688 nonn
%O A116688 1,4
%A A116688 _Emeric Deutsch_, Feb 23 2006