A182708 -id:A182708 - cp's OEIS frontend

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

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

Offset: 1

Omar E. Pol, Nov 28 2010, Nov 29 2010

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

			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.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jason Kimberley)
Omar E. Pol, Illustration: How to build the last section of the set of partitions (copy, paste and fill)
Omar E. Pol, Illustration of the shell model of partitions (2D view)

Cf. A000041, A135010, A138121, A138879, A138880, A182699, A182703, A182708, A182740, A182742, A182743.

Row sums of A183152.

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

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 *)

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

More terms from Alois P. Heinz, Dec 01 2010

A182707 Sum of the parts of all partitions of n-1 plus the sum of the emergent parts of the partitions of n.

Original entry on oeis.org

0, 1, 4, 11, 23, 46, 80, 138, 221, 351, 529, 801, 1161, 1685, 2380, 3355, 4624, 6375, 8623, 11658, 15538, 20664, 27163, 35660, 46330, 60082, 77288, 99197, 126418, 160802, 203246, 256381, 321700, 402781, 501962, 624332, 773235, 955776, 1177076, 1446762, 1772308

Offset: 1

Omar E. Pol, Nov 28 2010

For more information about the emergent parts of the partitions of n see A182699 and A182709.

			For n = 6 the partitions of 6-1=5 are (5);(3+2);(4+1);(2+2+1);(3+1+1);(2+1+1+1);(1+1+1+1+1) and the sum of the parts give 35, the same as 5*7. By other hand the emergent parts of the partitions of 6 are (2+2);(4);(3) and the sum give 11, so a(6) = 35+11 = 46.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Jason Kimberley)

Cf. A000041, A046746, A066186, A135010, A138121, A182699, A182708, A182709.

a(n) = A066186(n) - A046746(n) = A066186(n-1) + A182709(n).

a(n) ~ exp(Pi*sqrt(2*n/3)) / (4*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6)))/sqrt(n)). - Vaclav Kotesovec, Jan 03 2019, extended Jul 06 2019

A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 2, 4, 2, 5, 2, 2, 3, 6, 2, 2, 3, 7, 2, 2, 2, 2, 3, 4, 8, 2, 2, 2, 2, 3, 3, 4, 9, 2, 2, 2, 2, 2, 2, 2, 3, 3, 4, 5, 10, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 4, 5, 11, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 6, 12, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 4, 4, 5, 6, 13

Offset: 1

Omar E. Pol, Oct 21 2011

For n >= 2, row n lists the parts of the head of the last section of the set of partitions of n, except the emergent parts.

Also 1 together with the integers > 1 of A196931.

			Written as a triangle:
1,
2,
3,
2,4,
2,5,
2,2,3,6
2,2,3,7,
2,2,2,2,3,4,8,
2,2,2,2,3,3,4,9,
2,2,2,2,2,2,2,3,3,4,5,10,
2,2,2,2,2,2,2,2,3,3,3,4,5,11,
2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,4,4,5,6,12,
2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,5,6,13,
...
Row n has length A002865(n), n >= 2. The sum of row n is A182708(n), n >= 2. The number of 2's in row n is A002865(n-2), n >= 4. Right border of triangle gives A000027.

Alois P. Heinz, Rows n = 1..33, flattened

Where records occur give A000041.

Cf. A002865, A046746, A135010, A138121, A141285, A182699, A183152, A193827, A196931.

p:= (f, g)-> zip((x, y)->x+y, f, g, 0):
b:= proc(n, i) option remember; local g, j, r;
      if n=0 then [1] elif i<2 then [0]
    else r:= b(n, i-1);
         for j to n/i do g:= b(n-i*j, i-1);
           r:= p(p(r, [0$i, g[1]]), subsop(1=0, g));
         od; r
      fi
    end:
T:= proc(n) local l; l:= b(n$2);
      `if`(n=1, 1, seq(i$l[i+1], i=2..nops(l)-1))
    end:
seq(T(n), n=1..16);  # Alois P. Heinz, May 30 2013

p[f_, g_] := Plus @@ PadRight[{f, g}]; b[n_, i_] := b[n, i] = Module[{ g, j, r}, Which[n == 0, {1}, i<2, {0}, True, r = b[n, i-1]; For[j = 1, j <= n/i, j++, g = b[n-i*j, i-1]; r = p[p[r, Append[Array[0&, i], g // First]], ReplacePart[g, 1 -> 0]]]; r]]; T[n_] := Module[{l}, l = b[n, n]; If[n == 1, {1}, Table[Array[i&, l[[i+1]]], {i, 2, Length[l]-1}] // Flatten]]; Table[T[n], {n, 1, 16}] // Flatten (* Jean-François Alcover, Jan 30 2014, after Alois P. Heinz *)

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

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A182707 Sum of the parts of all partitions of n-1 plus the sum of the emergent parts of the partitions of n.

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A196930 Triangle read by rows in which row n lists in nondecreasing order the smallest part of every partition of n that do not contain 1 as a part, with a(1) = 1.

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