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A117455 Sum of the differences between the largest part and smallest part over all partitions of n into distinct parts.

%I A117455 #16 Jul 11 2016 11:20:06
%S A117455 0,0,1,2,4,8,12,19,27,41,54,76,99,133,171,223,279,357,443,554,682,841,
%T A117455 1022,1247,1504,1814,2174,2603,3092,3676,4346,5127,6030,7076,8275,
%U A117455 9669,11254,13078,15167,17556,20270,23377,26899,30902,35448,40592,46403
%N A117455 Sum of the differences between the largest part and smallest part over all partitions of n into distinct parts.
%C A117455 a(n) = sum(k*A117454(n,k), k=0..n-2).
%C A117455 a(n) = A005895(n)-A092265(n). - _Alois P. Heinz_, Jul 06 2012
%H A117455 Alois P. Heinz, <a href="/A117455/b117455.txt">Table of n, a(n) for n = 1..10000</a>
%F A117455 G.f.: sum(x^(i(i+1)/2)*sum(1/(1-x^j), j=1..i-1)/product(1-x^j, j=1..i), i=1..infinity) (obtained by taking the derivative with respect to t of the g.f. G(t,x) of A117454 and letting t=1).
%e A117455 a(7)=12 because the partitions of 7 into distinct parts are [7], [6,1], [5,2], [4,3] and [4,2,1] and (7-7)+(6-1)+(5-2)+(4-3)+(4-1)=12.
%p A117455 g:=sum(x^(i*(i+1)/2)*sum(1/(1-x^j),j=1..i-1)/product(1-x^j,j=1..i),i=1..15): gser:=series(g,x=0,55): seq(coeff(gser,x^n), n=1..50);
%p A117455 # second Maple program:
%p A117455 b:= proc(n, i) option remember;
%p A117455       `if`(i=n, n, 0)+`if`(i>0, b(n, i-1)+
%p A117455       `if`(i<n, b(n-i, i-1), 0), 0)
%p A117455     end:
%p A117455 g:= proc(n, i) option remember;
%p A117455       `if`(i=n, n, 0)+`if`(i<n, g(n, i+1)+g(n-i, i+1), 0)
%p A117455     end:
%p A117455 a:= n-> g(n, 1) -b(n, n):
%p A117455 seq(a(n), n=1..60);  # _Alois P. Heinz_, Jul 06 2012
%t A117455 b[n_, i_] := b[n, i] = If[i==n, n, 0] + If[i>0, b[n, i-1] + If[i<n, b[n-i, i-1], 0], 0]; g[n_, i_] := g[n, i] = If[i==n, n, 0] + If[i<n, g[n, i+1] + g[n-i, i+1], 0]; a[n_] := g[n, 1] - b[n, n]; Table[a[n], {n, 1, 60}] (* _Jean-François Alcover_, Mar 24 2015, after _Alois P. Heinz_ *)
%Y A117455 Cf. A005895, A092265, A117454.
%K A117455 nonn
%O A117455 1,4
%A A117455 _Emeric Deutsch_, Mar 18 2006