cp's OEIS Frontend

A182709 Sum of the emergent parts of the partitions of n.

%I A182709 #51 Jul 05 2019 18:00:00
%S A182709 0,0,0,2,3,11,14,33,45,81,109,185,237,372,490,715,928,1326,1693,2348,
%T A182709 2998,4032,5119,6795,8530,11132,13952,17927,22314,28417,35126,44279,
%U A182709 54532,68062,83422,103427,126063,155207,188506,230547,278788,339223,408482
%N A182709 Sum of the emergent parts of the partitions of n.
%C A182709 Here the "emergent parts" of the partitions of n are defined to be the parts (with multiplicity) of all the partitions that do not contain "1" as a part, removed by one copy of the smallest part of every partition. Note that these parts are located in the head of the last section of the set of partitions of n. For more information see A182699.
%C A182709 Also total sum of parts of the regions that do not contain 1 as a part in the last section of the set of partitions of n (Cf. A083751, A187219). - _Omar E. Pol_, Mar 04 2012
%H A182709 Vaclav Kotesovec, <a href="/A182709/b182709.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Jason Kimberley)
%H A182709 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpanxt.jpg">Illustration: How to build the last section of the set of partitions (copy, paste and fill)</a>
%H A182709 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa2dt.jpg">Illustration of the shell model of partitions (2D view)</a>
%F A182709 a(n) = A138880(n) - A182708(n).
%F A182709 a(n) = A066186(n) - A066186(n-1) - A046746(n) = A138879(n) - A046746(n). - _Omar E. Pol_, Aug 01 2013
%F A182709 a(n) ~ Pi * exp(Pi*sqrt(2*n/3)) / (12*sqrt(2*n)) * (1 - (3*sqrt(3/2)/Pi + 13*Pi/(24*sqrt(6)))/sqrt(n)). - _Vaclav Kotesovec_, Jan 03 2019, extended Jul 05 2019
%e A182709 For n=7 the partitions of 7 that do not contain "1" as a part are
%e A182709 7
%e A182709 4 + 3
%e A182709 5 + 2
%e A182709 3 + 2 + 2
%e A182709 Then remove one copy of the smallest part of every partition. The rest are the emergent parts:
%e A182709 .,
%e A182709 4, .
%e A182709 5, .
%e A182709 3, 2, .
%e A182709 The sum of these parts is 4 + 5 + 3 + 2 = 14, so a(7)=14.
%e A182709 For n=10 the illustration in the link shows the location of the emergent parts (colored yellow and green) and the location of the filler parts (colored blue) in the last section of the set of partitions of 10.
%p A182709 b:= proc(n, i) option remember;
%p A182709       if n<0 then 0
%p A182709     elif n=0 then 1
%p A182709     elif i<2 then 0
%p A182709     else b(n, i-1) +b(n-i, i)
%p A182709       fi
%p A182709     end:
%p A182709 c:= proc(n, i, k) option remember;
%p A182709       if n<0 then 0
%p A182709     elif n=0 then k
%p A182709     elif i<2 then 0
%p A182709     else c(n, i-1, k) +c(n-i, i, i)
%p A182709       fi
%p A182709     end:
%p A182709 a:= n-> n*b(n, n) - c(n, n, 0):
%p A182709 seq(a(n), n=1..40);  #  _Alois P. Heinz_, Dec 01 2010
%t A182709 f[n_]:=Total[Flatten[Most/@Select[IntegerPartitions[n],!MemberQ[#,1]&]]]; Table[f[i],{i,50}] (* _Harvey P. Dale_, Dec 28 2010 *)
%t A182709 b[n_, i_] := b[n, i] = Which[n<0, 0, n==0, 1, i<2, 0, True, b[n, i-1] + b[n - i, i]]; c[n_, i_, k_] := c[n, i, k] = Which[n<0, 0, n==0, k, i<2, 0, True, c[n, i-1, k] + c[n-i, i, i]]; a[n_] := n*b[n, n] - c[n, n, 0]; Table[a[n], {n, 1, 40}] (* _Jean-François Alcover_, Oct 08 2015, after _Alois P. Heinz_ *)
%Y A182709 Cf. A000041, A135010, A138121, A138879, A138880, A182699, A182703, A182708, A182740, A182742, A182743.
%Y A182709 Row sums of A183152.
%K A182709 nonn,easy
%O A182709 1,4
%A A182709 _Omar E. Pol_, Nov 28 2010, Nov 29 2010
%E A182709 More terms from _Alois P. Heinz_, Dec 01 2010