A138880 -id:A138880 - cp's OEIS frontend

A206562 Triangle read by rows: T(n,k) = sum of all parts >= k in the last section of the set of partitions of n.

1, 3, 2, 5, 3, 3, 11, 8, 4, 4, 15, 10, 8, 5, 5, 31, 24, 16, 10, 6, 6, 39, 28, 22, 16, 12, 7, 7, 71, 56, 40, 31, 19, 14, 8, 8, 94, 72, 58, 40, 32, 22, 16, 9, 9, 150, 120, 90, 72, 52, 37, 25, 18, 10, 10, 196, 154, 124, 94, 74, 54, 42, 28, 20, 11, 11

Offset: 1

Omar E. Pol, Feb 15 2012

			Triangle begins:
1;
3,   2;
5,   3,  3;
11,  8,  4,  4;
15, 10,  8,  5,  5;
31, 24, 16, 10,  6,  6;
39, 28, 22, 16, 12,  7,  7;
71, 56, 40, 31, 19, 14,  8,  8;
94, 72, 58, 40, 32, 22, 16,  9,  9;

Alois P. Heinz, Rows n = 1..141, flattened

Columns 1-2 give A138879, A138880. Diagonal is A000027.

Cf. A135010, A138121, A182703, A206561, A207031, A207032.

A194552 Sum of all parts > 1 of all partitions of n.

Original entry on oeis.org

0, 2, 5, 13, 23, 47, 75, 131, 203, 323, 477, 729, 1041, 1517, 2132, 3012, 4134, 5718, 7713, 10453, 13918, 18538, 24357, 32037, 41612, 54040, 69538, 89362, 113925, 145095, 183473, 231697, 290899, 364577, 454632, 566016, 701436, 867800, 1069430, 1315550, 1612595

Offset: 1

Omar E. Pol, Dec 11 2011

nonn

Also the total number of missing parts in the partitions of n. A missing part of a partition of n is any number from 1 to n not occurring as a part. For example for n = 3, 1,2 are missing from 3; 3 is missing from 2+1, and 2,3 are missing from 1+1+1, for a total of a(3) = 5. - George Beck, Oct 23 2014

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Partial sums of A138880.

Cf. A096541, A135010, A138121, A138135.

b:= proc(n, i) option remember; local h, t;
      if n<0 or i<1 then [0, 0]
    elif n=0 or i=1 then [1, 0]
    else h:= b(n, i-1); t:= b(n-i, i);
         [h[1]+t[1], h[2]+t[2] +t[1]*i]
      fi
    end:
a:= n-> b(n, n)[2]:
seq(a(n), n=1..50); # Alois P. Heinz, Dec 14 2011

a[n_] := n PartitionsP[n] -Total@Table[PartitionsP[k], {k, 0, n - 1}]; a /@ Range[40] (* George Beck, Oct 23 2014 *)

a(n) = A066186(n) - A000070(n-1).
a(n) = n * A000041(n) - A000070(n-1). - George Beck, Oct 24 2014
G.f.: (x/(1 - x)) * (d/dx) Product_{k>=2} 1/(1 - x^k). - Ilya Gutkovskiy, Mar 06 2021

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

Original entry on oeis.org

0, -2, 1, -7, 3, -21, 7, -49, 23, -97, 57, -195, 117, -359, 256, -624, 498, -1086, 909, -1831, 1634, -2986, 2833, -4847, 4728, -7700, 7798, -12026, 12537, -18633, 19745, -28479, 30723, -42955, 47100, -64284, 71136, -95228, 106402, -139718, 157327, -203495

Offset: 1

Omar E. Pol, Jan 31 2012

sign

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.

			For n = 6 the illustration of the three views of the shell model with 6 shells shows an imbalance (see below):
------------------------------------------------------
Partitions                Tree             Table 1.0
of 6.                    A194805            A135010
------------------------------------------------------
6                   6                     6 . . . . .
3+3                   3                   3 . . 3 . .
4+2                     4                 4 . . . 2 .
2+2+2                     2               2 . 2 . 2 .
5+1                         1   5         5 . . . . 1
3+2+1                       1 3           3 . . 2 . 1
4+1+1                   4   1             4 . . . 1 1
2+2+1+1                   2 1             2 . 2 . 1 1
3+1+1+1                     1 3           3 . . 1 1 1
2+1+1+1+1                 2 1             2 . 1 1 1 1
1+1+1+1+1+1                 1             1 1 1 1 1 1
------------------------------------------------------
.
.                   6 3 4 2 1 3 5
.     Table 2.0     . . . . 1 . .     Table 2.1
.      A182982      . . . 2 1 . .      A182983
.                   . 3 . . 1 2 .
.                   . . 2 2 1 . .
.                   . . . . 1
------------------------------------------------------
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.

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A000041, A002865, A135010, A138121, A138880, A141285, A182710, A182742, A182743, A182746, A182747, A182982, A182983, A182994, A182995, A194796, A194805.

with(combinat):
a:= proc(n) option remember;
      n *(-1)^n *(numbpart(n-1)-numbpart(n)) +a(n-1)
    end: a(0):=0:
seq(a(n), n=1..50); # Alois P. Heinz, Apr 04 2012

a[n_] := Sum[(-1)^(k-1)*k*(PartitionsP[k] - PartitionsP[k-1]), {k, 1, n}]; Array[a, 50] (* Jean-François Alcover, Dec 09 2016 *)

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.
a(n) = b(1)-b(2)+b(3)-b(4)+b(5)-b(6)...+-b(n), where b(n) = A138880(n).
a(n) ~ -(-1)^n * Pi * sqrt(2) * exp(Pi*sqrt(2*n/3)) / (48*sqrt(n)). - Vaclav Kotesovec, Oct 09 2018

A194804 Sum of parts that are visible in one of the three views of the shell model of partitions version "tree" with n shells.

Original entry on oeis.org

0, 1, 4, 8, 15, 23, 40, 59, 92, 137, 202, 285, 418, 577, 802, 1106, 1511, 2019, 2724, 3598, 4755, 6226, 8107, 10462, 13523, 17280, 22029, 27953, 35350, 44416, 55763, 69579, 86634, 107459, 132914, 163768, 201475, 246841, 301822, 368033, 447790, 543206

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

For the number of parts see A194803. For more information about the shell model see A135010 and A194805.

			Illustration of one of the three views with seven shells:
.
.        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
.                   . . . . 1 . . . .
.                   . . . 2 1 . . . .
.      Table 2.0    . 3 . . 1 2 . . .    Table 2.1
.                   . . 2 2 1 . . 3 .
.                   . . . . 1 2 2 . .
.                           1 . . . .
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
The sum of parts that are visible is 1+1+1+1+1+1+1+2+2+2+2+2+2+2+3+3+3+3+4+4+5+6+7 = 59, so a(7) = 59. Using the formula we have a(7) = 7+24+28 = 59.

Cf. A002865, A066186, A135010, A138121, A138880, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194803, A194805.

a(n) = n + A138880(n-1) + A138880(n), if n >= 2.

A182737 Sum of parts in all partitions of 2n+1 that do not contain 1 as a part.

Original entry on oeis.org

0, 3, 10, 28, 72, 154, 312, 615, 1122, 1995, 3465, 5819, 9575, 15498, 24563, 38378, 59202, 90055, 135420, 201630, 297045, 433741, 628155, 902212, 1286348, 1821567, 2562126, 3581655, 4977867, 6879400, 9457318, 12936609, 17610320, 23863323, 32196090

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

Bisection (odd part) of A138880.

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Cf. A135010, A138121, A138880, A182736, A182743.

b:= proc(n,i) option remember; local p,q;
      if n<0 then [0,0]
    elif n=0 then [1,0]
    elif i<2 then [0,0]
    else p, q:= b(n,i-1), b(n-i,i);
        [p[1]+q[1], p[2]+q[2]+q[1]*i]
      fi
    end:
a:= n-> b(2*n+1,2*n+1)[2]:
seq(a(n), n=0..34); # Alois P. Heinz, Dec 03 2010

b[n_, i_] := b[n, i] = Module[{p, q}, Which[n<0, {0, 0}, n == 0, {1, 0}, i<2, {0, 0}, True, {p, q} = {b[n, i-1], b[n-i, i]}; {p[[1]] + q[[1]], p[[2]] + q[[2]] + q[[1]]*i}]]; a[n_] := b[2*n + 1, 2*n+1][[2]]; Table[ a[n], {n, 0, 34}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

a(n) = A005408(n)*A182747(n).

More terms from Alois P. Heinz, Dec 03 2010

A182711 Triangle read by rows in which row n lists the parts > 1 of the last section of the set of partitions of n in an order similar to A138136 but in this case the partitions with the least number of parts are listed first.

Original entry on oeis.org

0, 2, 3, 4, 2, 2, 5, 3, 2, 6, 4, 2, 3, 3, 2, 2, 2, 7, 5, 2, 4, 3, 3, 2, 2, 8, 6, 2, 5, 3, 4, 4, 4, 2, 2, 3, 3, 2, 2, 2, 2, 2, 9, 7, 2, 6, 3, 5, 4, 5, 2, 2, 4, 3, 2, 3, 3, 3, 3, 2, 2, 2, 10, 8, 2, 7, 3, 6, 4, 5, 5, 6, 2, 2, 5, 3, 2, 4, 4, 2, 4, 3, 3, 4, 2, 2, 2, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Nov 29 2010

In this sequence a(68)=5 but in A138136 a(68)=6. See the 8th term in row 10 of triangle.

			0,
2,
3,
4,2,2,
5,3,2,
6,4,2,3,3,2,2,2,
7,5,2,4,3,3,2,2,
8,6,2,5,3,4,4,4,2,2,3,3,2,2,2,2,2,
9,7,2,6,3,5,4,5,2,2,4,3,2,3,3,3,3,2,2,2,
10,8,2,7,3,6,4,5,5,6,2,2,5,3,2,4,4,2,4,3,3,4,2,2,2,3,3,2,2,2,2,2,2,2

Cf. A135010, A138121, A138136, A138151, A182710.

Row sums give A138880.

A139094 Largest part of the n-th row in the integrated diagram of the shell model of partitions.

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, May 26 2008

nonn

Omar E. Pol, A shell model of partitions
Omar E. Pol, Illustration of initial terms

Cf. A138135, A138136, A138137, A138138, A138151, A138879, A138880, A139100.

A194450 Vertex number of a rectangular spiral which contains exactly between its edges the successive shells of the partitions of the positive integers.

Original entry on oeis.org

0, 1, 2, 4, 6, 9, 12, 17, 21, 28, 33, 44, 50, 65, 72, 94, 102, 132, 141, 183, 193, 249, 260, 337, 349, 450, 463, 598, 612, 788, 803, 1034, 1050, 1347, 1364, 1749, 1767, 2257, 2276, 2903, 2923, 3715, 3736, 4738, 4760, 6015, 6038, 7613, 7637, 9595

Offset: 0

Omar E. Pol, Nov 01 2011

nonn

First differences give A194451, the length of the edges of the spiral. For more information see A135010 and A138121.

Cf. A000041, A000217, A006128, A026905, A066186, A135010, A138121, A138879, A138880, A182703, A182994, A182995, A194451.

a(2n-1) = A026905(n) + A000217(n) - n, if n >= 1.

a(2n) = A026905(n) + A000217(n), if n >= 1.

A228816 Sum of all parts of all partitions of n that contain 1 as a part.

Original entry on oeis.org

1, 2, 6, 12, 25, 42, 77, 120, 198, 300, 462, 672, 1001, 1414, 2025, 2816, 3927, 5346, 7315, 9800, 13167, 17424, 23046, 30120, 39375, 50908, 65772, 84280, 107822, 136950, 173724, 218944, 275517, 344862, 430850, 535788, 665149, 822206, 1014585, 1247400

Offset: 1

Omar E. Pol, Sep 23 2013

nonn

Column 1 of A182701.

Cf. A000041, A000070, A066186, A138880, A194552, A228823.

Table[Total[Flatten[Select[IntegerPartitions[n],MemberQ[#,1]&]]],{n,40}] (* Harvey P. Dale, Sep 27 2015 *)

a(n) = n*A000041(n-1).
a(n) = A066186(n-1) + A000041(n-1), n >= 2.
a(n) = A194552(n-1) + A000070(n-1), n >= 2.

cp's OEIS Frontend

A206562 Triangle read by rows: T(n,k) = sum of all parts >= k in the last section of the set of partitions of n.

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A194552 Sum of all parts > 1 of all partitions of n.

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

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A194804 Sum of parts that are visible in one of the three views of the shell model of partitions version "tree" with n shells.

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A182737 Sum of parts in all partitions of 2n+1 that do not contain 1 as a part.

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A182711 Triangle read by rows in which row n lists the parts > 1 of the last section of the set of partitions of n in an order similar to A138136 but in this case the partitions with the least number of parts are listed first.

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A139094 Largest part of the n-th row in the integrated diagram of the shell model of partitions.

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A194450 Vertex number of a rectangular spiral which contains exactly between its edges the successive shells of the partitions of the positive integers.

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A228816 Sum of all parts of all partitions of n that contain 1 as a part.

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Mathematica

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