cp's OEIS Frontend

A194809 Imbalance of the sum of largest parts of all partitions of n.

%I A194809 #14 Oct 16 2018 11:23:54
%S A194809 0,-2,1,-5,3,-12,7,-25,17,-47,36,-88,69,-155,133,-262,240,-439,415,
%T A194809 -717,705,-1142,1165,-1803,1874,-2797,2975,-4276,4632,-6478,7094,
%U A194809 -9698,10741,-14355,16059,-21079,23719,-30670,34716,-44243,50315,-63372
%N A194809 Imbalance of the sum of largest parts of all partitions of n.
%C A194809 Consider the three-dimensional structure of the shell model of partitions version "tree". Note that only the larges parts > 1 produce the imbalance. Note that every column where is located a largest part contains largest parts of the same size, thesame as a periodic table (see example). For more information see A135010.
%F A194809 a(n) = Sum_{k=2..n} (-1)^(k-1)*A138137(k), n >= 2.
%e A194809 For n = 6 the illustration of the shell model with 6 shells shows an imbalance of largest parts (see below):
%e A194809 ------------------------------------------------------
%e A194809 Partitions                Tree             Table 1.0
%e A194809 of 6.                    A194805            A135010
%e A194809 ------------------------------------------------------
%e A194809 6                   6                     6 . . . . .
%e A194809 3+3                   3                   3 . . 3 . .
%e A194809 4+2                     4                 4 . . . 2 .
%e A194809 2+2+2                     2               2 . 2 . 2 .
%e A194809 5+1                         1   5         5 . . . . 1
%e A194809 3+2+1                       1 3           3 . . 2 . 1
%e A194809 4+1+1                   4   1             4 . . . 1 1
%e A194809 2+2+1+1                   2 1             2 . 2 . 1 1
%e A194809 3+1+1+1                     1 3           3 . . 1 1 1
%e A194809 2+1+1+1+1                 2 1             2 . 1 1 1 1
%e A194809 1+1+1+1+1+1                 1             1 1 1 1 1 1
%e A194809 ------------------------------------------------------
%e A194809 The sum of largest parts > 1 on the left hand side is 23 and the sum of largest parts > 1 on the right hand side is 11, so a(6) = -23 + 11 = -12. On the other hand for n = 6 we have that 0 together with the first n-1 terms > 1 of A138137 are 0, 2, 3, 6, 8, 15 so a(6) = 0-2+3-6+8-15 = -12.
%Y A194809 Cf. A135010, A138121, A138137, A141285, A194795-A194797, A194805.
%K A194809 sign
%O A194809 1,2
%A A194809 _Omar E. Pol_, Feb 02 2012