A182743 -id:A182743 - cp's OEIS frontend

A183009 a(n) = 24np(n) = 24nA000041(n).

24, 96, 216, 480, 840, 1584, 2520, 4224, 6480, 10080, 14784, 22176, 31512, 45360, 63360, 88704, 121176, 166320, 223440, 300960, 399168, 529056, 692760, 907200, 1174800, 1520064, 1950480, 2498496, 3177240, 4034880, 5090448, 6412032

Offset: 1

Omar E. Pol, Jan 22 2011

a(n) is also the sum of the partition number of n and the "trace" Tr(n) of A183011. a(n) = p(n) + Tr(n).

a(n) is also the number of "sectors" or "half-periods" in all partitions of n in some versions of the shell model of partitions of A135010.

			The number of partitions of 6 is p(6) = A000041(6) = 11, so a(6) = 24*6*11 = 1584.
Also the trace Tr(6) = A183011(6) = 1573, so a(6) = p(6) + Tr(6) = 11 + 1573 = 1584.

Partial sums of A183006. Column 24 of A182728.

Cf. A000041, A000796, A008606, A018253, A066186, A135010, A182742, A182743, A183008, A183010, A183011.

Table[24n*PartitionsP[n],{n,40}] (* Harvey P. Dale, Mar 07 2019 *)

a(n) = A008606(n)*A000041(n) = 24*A066186(n) = n*A183008(n).
a(n) = p(n) + Tr(n) = A000041(n) + A183011(n).
a(n) = 12*M_2(n) = 24*spt(n) + 12*N_2(n) = 12*A220909(n) = 24*A092269(n) + 12*A220908(n). - Omar E. Pol, Feb 17 2013

A182993 Number of parts of the n-th subshell of the head of the last section of the set of partitions of any odd integer >= 2n+1.

Original entry on oeis.org

1, 2, 5, 12, 21, 39, 73, 118, 198, 326, 510, 797, 1234, 1854, 2778, 4122, 6014, 8717, 12550, 17849, 25252, 35486, 49447, 68540, 94480, 129378, 176339, 239165, 322676, 433487, 579907, 772318, 1024691, 1354445, 1783504

Offset: 1

Omar E. Pol, Feb 06 2011

nonn

The last section of the set of partitions of 2n+1 contains n subshells.

Also first differences of A182735. - Omar E. Pol, Mar 03 2011

			a(5)=21 because the 5th subshell of the head of the last section of any odd integer >= 11 looks like this:
(11 . . . . . . . . . . )
( 6 . . . . . 5 . . . . )
( 7 . . . . . . 4 . . . )
( 8 . . . . . . . 3 . . )
( 4 . . . 4 . . . 3 . . )
( 5 . . . . 3 . . 3 . . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
.                  (2 . )
The subshell has 21 parts, so a(5)=21.

Cf. A135010, A138135, A182735, A182743, A182983, A182992, A182995.

a(n) = A138135(2n+1) - A138135(2n-1).

More terms from Omar E. Pol, Mar 03 2011

A194803 Number 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, 3, 5, 8, 11, 17, 23, 33, 46, 64, 86, 121, 161, 217, 291, 388, 507, 671, 870, 1131, 1458, 1872, 2383, 3042, 3840, 4841, 6076, 7605, 9460, 11765, 14544, 17950, 22073, 27077, 33092, 40395, 49113, 59611, 72162, 87185, 105035, 126366

Offset: 0

Omar E. Pol, Jan 27 2012

nonn

The physical model shows each part represented by an object, for example using a cube or a cuboid. In this case the small version of the model shows each part as a cube of side 1 which is labeled with the size of the part. On the same way the large version of the model shows each part as a cuboid of sides 1 x 1 x L where L is the size of the part. The cuboid is labeled with the level of the part. For the sum of parts see A194804. For more information about the shell model see A135010 and A194805.

			Illustration of one of the three views with seven shells:
1) Small version:
.
Level
1        A182732 <- 6 3 4 2 1 3 5 4 7 -> A182733
2                     3 2 2 1 2 2 3
3                         2 1 2
4                           1
5      Table 2.0            1            Table 2.1
6                           1
7                           1
.
.  A182742  A182982                   A182743  A182983
.  A182992  A182994                   A182993  A182995
.
2) Large version:
.
.                   . . . . 1 . . . .
.                   . . . 1 2 . . . .
.                   . 1 . . 2 1 . . .
.                   . . 1 2 2 . . 1 .
.                   . . . . 2 2 1 . .
.                   1 2 2 3 2 . . . .
.                           2 3 2 2 1
.
The large version shows the parts labeled with the level of the part where "the level of a part" is its position in the partition. In both versions there are 23 parts that are visible, so a(7) = 23. Also using the formula we have a(7) = 7+8+8 = 23.

Cf. A006128, A096541, A138135, A135010, A138121, A141285, A182732, A182733, A182742, A182743, A182982, A182983, A182992-A182995, A194804, A194805, A210979.

a(n) = n + A138135(n-1) + A138135(n), if n >= 2.

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.

A182735 Number of parts in all partitions of 2n+1 that do not contain 1 as a part.

Original entry on oeis.org

0, 1, 3, 8, 20, 41, 80, 153, 271, 469, 795, 1305, 2102, 3336, 5190, 7968, 12090, 18104, 26821, 39371, 57220, 82472, 117958, 167405, 235945, 330425, 459803, 636142, 875307, 1197983, 1631470, 2211377, 2983695, 4008386, 5362831, 7146335, 9486834, 12548085, 16538651

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

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

Cf. A135010, A138121, A182734, A182743, A182747. Bisection (odd part) of A138135.

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]]
      fi
    end:
a:= n-> b(2*n+1, 2*n+1)[2]:
seq(a(n), n=0..35);  # 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]]}]]; a[n_] := b[2*n+1, 2*n+1][[2]]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Oct 29 2015, after Alois P. Heinz *)

More terms from Alois P. Heinz, Dec 03 2010

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

A182744 Second column of the table A182742.

Original entry on oeis.org

2, 2, 3, 2, 3, 4, 2, 3, 3, 4, 5, 2, 3, 3, 4, 3, 4, 4, 5, 6, 2, 3, 4, 3, 4, 5, 3, 4, 5, 4, 5, 6, 7, 2

Offset: 1

Omar E. Pol, Nov 30 2010

The second largest part of the n-th partition of the table A182742.

Cf. A135010, A138121, A141285, A182730, A182732, A182742, A182743, A182744, A182746.

A182980 Version "mirror" of the shell model of partitions of A135010. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 1, 1, 1, 4, 2, 2, 1, 1, 1, 1, 1, 2, 3, 5, 1, 1, 1, 1, 1, 1, 1, 6, 3, 3, 2, 4, 2, 2, 2, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 5, 3, 4, 7, 8, 4, 4, 3, 5, 2, 6, 2, 3, 3, 2, 2, 4, 2, 2, 2, 2

Offset: 1

Omar E. Pol, Feb 02 2011

In n is odd then row n lists the numbers of row n of A135010. If n is even the row n lists the 1's of row n of A135010 and then row n lists the other numbers of row n of A135010 in reverse order.

			Triangle begins:
1,
1, 2,
1, 1, 3,
1, 1, 1, 4, 2, 2,
1, 1, 1, 1, 1, 2, 3, 5,
1, 1, 1, 1, 1, 1, 1, 6, 3, 3, 2, 4, 2, 2, 2,
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 2, 5, 3, 4, 7,

Cf. A135010, A141285, A182742, A182743, A182982, A182983, A182985.

A182717 Number of 2's in all partitions of 2n+1 that do not contain 1 as a part.

Original entry on oeis.org

0, 0, 1, 3, 7, 15, 29, 53, 94, 160, 265, 430, 683, 1066, 1640, 2487, 3725, 5519, 8092, 11752, 16922, 24167, 34254, 48213, 67409, 93661, 129378, 177720, 242841, 330172, 446772, 601810, 807153, 1078081, 1434250, 1900860, 2510097, 3303003, 4331767, 5662539

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

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

A182743. Bisection of A182712.

b:= proc(n,i) option remember; local r;
      if n<=0 or i<2 then 0
    elif i=2 then `if`(irem(n,2,'r')=0,r,0)
    else b(n,i-1) +b(n-i,i)
      fi
    end:
a:= n-> b(2*n+1, 2*n+1):
seq(a(n), n=0..45);  # Alois P. Heinz, Dec 03 2010

b[n_, i_] := b[n, i] = If[n <= 0 || i < 2, 0, If[i == 2, If[Mod[n, 2] == 0, Quotient[n, 2], 0], b[n, i-1] + b[n-i, i]]];
a[n_] := b[2n+1, 2n+1];
a /@ Range[0, 45] (* Jean-François Alcover, Nov 11 2020, after Alois P. Heinz *)
Table[Count[Flatten[Select[IntegerPartitions[2 n+1],FreeQ[#,1]&]],2],{n,0,40}] (* Harvey P. Dale, Jan 28 2022 *)

More terms from Alois P. Heinz, Dec 03 2010

cp's OEIS Frontend

A183009 a(n) = 24*n*p(n) = 24*n*A000041(n).

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A182993 Number of parts of the n-th subshell of the head of the last section of the set of partitions of any odd integer >= 2n+1.

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A194803 Number 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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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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A182735 Number of parts in all partitions of 2n+1 that do not contain 1 as a part.

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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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A182744 Second column of the table A182742.

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A182980 Version "mirror" of the shell model of partitions of A135010. Triangle read by rows: row n lists the parts of the last section of the set of partitions of n.

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A183009 a(n) = 24np(n) = 24nA000041(n).