cp's OEIS Frontend

A194797 Imbalance of the sum of parts of all partitions of n.

%I A194797 #77 Oct 09 2018 09:46:37
%S A194797 0,-2,1,-7,3,-21,7,-49,23,-97,57,-195,117,-359,256,-624,498,-1086,909,
%T A194797 -1831,1634,-2986,2833,-4847,4728,-7700,7798,-12026,12537,-18633,
%U A194797 19745,-28479,30723,-42955,47100,-64284,71136,-95228,106402,-139718,157327,-203495
%N A194797 Imbalance of the sum of parts of all partitions of n.
%C A194797 Consider the three-dimensional structure of the shell model of partitions, version "tree" (see example). Note that only the parts > 1 produce the imbalance. The 1's are located in the central columns therefore they do not produce the imbalance. Note that every column contains exactly the same parts. For more information see A135010.
%H A194797 Alois P. Heinz, <a href="/A194797/b194797.txt">Table of n, a(n) for n = 1..10000</a>
%F A194797 a(n) = Sum_{k=1..n} (-1)^(k-1)*k*(p(k)-p(k-1)), where p(k) is the number of partitions of k.
%F A194797 a(n) = b(1)-b(2)+b(3)-b(4)+b(5)-b(6)...+-b(n), where b(n) = A138880(n).
%F A194797 a(n) ~ -(-1)^n * Pi * sqrt(2) * exp(Pi*sqrt(2*n/3)) / (48*sqrt(n)). - _Vaclav Kotesovec_, Oct 09 2018
%e A194797 For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):
%e A194797 ------------------------------------------------------
%e A194797 Partitions                Tree             Table 1.0
%e A194797 of 6.                    A194805            A135010
%e A194797 ------------------------------------------------------
%e A194797 6                   6                     6 . . . . .
%e A194797 3+3                   3                   3 . . 3 . .
%e A194797 4+2                     4                 4 . . . 2 .
%e A194797 2+2+2                     2               2 . 2 . 2 .
%e A194797 5+1                         1   5         5 . . . . 1
%e A194797 3+2+1                       1 3           3 . . 2 . 1
%e A194797 4+1+1                   4   1             4 . . . 1 1
%e A194797 2+2+1+1                   2 1             2 . 2 . 1 1
%e A194797 3+1+1+1                     1 3           3 . . 1 1 1
%e A194797 2+1+1+1+1                 2 1             2 . 1 1 1 1
%e A194797 1+1+1+1+1+1                 1             1 1 1 1 1 1
%e A194797 ------------------------------------------------------
%e A194797 .
%e A194797 .                   6 3 4 2 1 3 5
%e A194797 .     Table 2.0     . . . . 1 . .     Table 2.1
%e A194797 .      A182982      . . . 2 1 . .      A182983
%e A194797 .                   . 3 . . 1 2 .
%e A194797 .                   . . 2 2 1 . .
%e A194797 .                   . . . . 1
%e A194797 ------------------------------------------------------
%e A194797 The sum of all parts > 1 on the left hand side is 34 and the sum of all parts > 1 on the right hand side is 13, so a(6) = -34 + 13 = -21. On the other hand for n = 6 the first n terms of A138880 are 0, 2, 3, 8, 10, 24 thus a(6) = 0-2+3-8+10-24 = -21.
%p A194797 with(combinat):
%p A194797 a:= proc(n) option remember;
%p A194797       n *(-1)^n *(numbpart(n-1)-numbpart(n)) +a(n-1)
%p A194797     end: a(0):=0:
%p A194797 seq(a(n), n=1..50); # _Alois P. Heinz_, Apr 04 2012
%t A194797 a[n_] := Sum[(-1)^(k-1)*k*(PartitionsP[k] - PartitionsP[k-1]), {k, 1, n}]; Array[a, 50] (* _Jean-François Alcover_, Dec 09 2016 *)
%Y A194797 Cf. A000041, A002865, A135010, A138121, A138880, A141285, A182710, A182742, A182743, A182746, A182747, A182982, A182983, A182994, A182995, A194796, A194805.
%K A194797 sign
%O A194797 1,2
%A A194797 _Omar E. Pol_, Jan 31 2012