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 A182709 Sum of the emergent parts of the partitions of n. 23
 0, 0, 0, 2, 3, 11, 14, 33, 45, 81, 109, 185, 237, 372, 490, 715, 928, 1326, 1693, 2348, 2998, 4032, 5119, 6795, 8530, 11132, 13952, 17927, 22314, 28417, 35126, 44279, 54532, 68062, 83422, 103427, 126063, 155207, 188506, 230547, 278788, 339223, 408482 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,4 COMMENTS 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. 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 LINKS Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jason Kimberley) FORMULA a(n) = A138880(n) - A182708(n). a(n) = A066186(n) - A066186(n-1) - A046746(n) = A138879(n) - A046746(n). - Omar E. Pol, Aug 01 2013 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 EXAMPLE For n=7 the partitions of 7 that do not contain "1" as a part are 7 4 + 3 5 + 2 3 + 2 + 2 Then remove one copy of the smallest part of every partition. The rest are the emergent parts: ., 4, . 5, . 3, 2, . The sum of these parts is 4 + 5 + 3 + 2 = 14, so a(7)=14. 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. MAPLE b:= proc(n, i) option remember;       if n<0 then 0     elif n=0 then 1     elif i<2 then 0     else b(n, i-1) +b(n-i, i)       fi     end: c:= proc(n, i, k) option remember;       if n<0 then 0     elif n=0 then k     elif i<2 then 0     else c(n, i-1, k) +c(n-i, i, i)       fi     end: a:= n-> n*b(n, n) - c(n, n, 0): seq(a(n), n=1..40);  #  Alois P. Heinz, Dec 01 2010 MATHEMATICA f[n_]:=Total[Flatten[Most/@Select[IntegerPartitions[n], !MemberQ[#, 1]&]]]; Table[f[i], {i, 50}] (* Harvey P. Dale, Dec 28 2010 *) 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 *) CROSSREFS Cf. A000041, A135010, A138121, A138879, A138880, A182699, A182703, A182708, A182740, A182742, A182743. Row sums of A183152. Sequence in context: A042665 A282235 A242821 * A321766 A041029 A158353 Adjacent sequences:  A182706 A182707 A182708 * A182710 A182711 A182712 KEYWORD nonn,easy AUTHOR Omar E. Pol, Nov 28 2010, Nov 29 2010 EXTENSIONS More terms from Alois P. Heinz, Dec 01 2010 STATUS approved

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Last modified July 4 02:51 EDT 2020. Contains 335436 sequences. (Running on oeis4.)