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%I #4 Oct 14 2014 19:07:44
%S 0,0,1,1,3,4,7,8,14,17,24,30,40,50,67,79,101,126,153,186,231,276,332,
%T 399,477,567,677,795,938,1111,1294,1512,1773,2058,2392,2775,3204,3701,
%U 4272,4904,5630,6467,7387,8442,9651,10980,12491,14202,16109,18260,20680
%N The difference between the largest part and the smallest part summed over all those partitions of n in which every integer from the smallest part to the largest part occurs.
%C a(n)=sum(k*A117470(n,k),k>=0).
%F G.f.=sum(x^j*product(1+x^i, i=1..j-1)sum(x^i/(1+x^i), i=1..j-1)/(1-x^j), j=1..infinity) (obtained by taking the derivative with respect to t of the g.f. G(t,x) of A117470 and setting t=1).
%e a(6)=4 because the 7 (=A034296(6) ) partitions of 6 in which every integer from the smallest part to the largest part occurs are [6],[3,3],[3,2,1],[2,2,2],[2,2,1,1],[2,1,1,1,1],[1,1,1,1,1,1] and (6-6)+(3-3)+(3-1)+(2-2)+(2-1)+(2-1)+(1-1)=4.
%p g:=sum(x^j*product(1+x^i,i=1..j-1)*sum(x^i/(1+x^i),i=1..j-1)/(1-x^j),j=1..65): gser:=series(g,x=0,60): seq(coeff(gser,x,n),n=1..57);
%t Table[Total[Max[#]-Min[#]&/@Select[IntegerPartitions[n],Max[Abs[ Differences[ #]]]<2&]],{n,60}] (* _Harvey P. Dale_, Oct 14 2014 *)
%Y Cf. A034296, A117470.
%K nonn
%O 1,5
%A _Emeric Deutsch_, Mar 20 2006
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