A138135 -id:A138135 - cp's OEIS frontend

A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

0, 1, 2, 5, 7, 16, 20, 39, 52, 86, 113, 184, 232, 353, 462, 661, 851, 1202, 1526, 2098, 2670, 3565, 4514, 5967, 7473, 9715, 12162, 15583, 19373, 24625, 30410, 38274, 47112, 58725, 71951, 89129, 108599, 133612, 162259, 198346, 239825, 291718, 351269, 425102

Offset: 1

Omar E. Pol, Feb 20 2012

nonn

			For n = 7 the last section of the set of partitions of 7 looks like this:
.
.        (. . . . . . 7)
.        (. . . 4 . . 3)
.        (. . . . 5 . 2)
.        (. . 3 . 2 . 2)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.                    (1)
.
The sum of all parts = 7+4+3+5+2+3+2+2+1*11 = 39, on the other hand the total number of parts is 1+2+2+3+1*11 = 19, so a(7) = 39 - 19 = 20. Note that the number of dots in the picture is also equal to a(7) = 6+5+5+4 = 20.

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

Row sums of triangle A207034. Partial sums give A196087.

Cf. A006128, A066186, A135010, A138121, A138135, A138137, A138879, A138880, A187219, A194548, A207038.

b:= proc(n, i) option remember; local f, g;
      if n=0 then [1, 0]
    elif i<2 then [0, 0]
    elif i>n then b(n, i-1)
    else f:= b(n, i-1); g:= b(n-i, i);
         [f[1]+g[1], f[2]+g[2] +g[1]*(i-1)]
      fi
    end:
a:= n-> b(n, n)[2]:
seq (a(n), n=1..50);  # Alois P. Heinz, Feb 20 2012

b[n_, i_] := b[n, i] = Module[{f, g}, Which[n==0, {1, 0}, i<2, {0, 0}, i>n , b[n, i-1], True, f = b[n, i-1]; g = b[n-i, i]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]*(i-1)}]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Sep 13 2015, after Alois P. Heinz *)

a(n) = A138879(n) - A138137(n) = A138880(n) - A138135(n). - Omar E. Pol, Apr 21 2012

G.f.: Sum_{k>=1} x^(2*k)/(1 - x^k)^2 / Product_{j>=2} (1 - x^j). - Ilya Gutkovskiy, Mar 05 2021

More terms from Alois P. Heinz, Feb 20 2012

A182734 Number of parts in all partitions of 2n that do not contain 1 as a part.

Original entry on oeis.org

0, 1, 3, 8, 17, 34, 68, 123, 219, 382, 642, 1055, 1713, 2713, 4241, 6545, 9950, 14953, 22255, 32752, 47774, 69104, 99114, 141094, 199489, 280096, 390836, 542170, 747793, 1025912, 1400425, 1902267, 2572095, 3462556, 4641516, 6196830, 8241460, 10919755, 14416885

Offset: 0

Omar E. Pol, Dec 03 2010

nonn

Essentially this is a bisection (even part) of A138135.

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

Cf. A135010, A138121, A138135, A182742.

b:= proc(n,i) option remember; local p,q;
      if n<0 then [0,0]
    elif n=0 then [1,0]
    elif i=1 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, 2*n)[2]:
seq(a(n), n=0..35);  # Alois P. Heinz, Dec 03 2010

Table[Length[Flatten[DeleteCases[IntegerPartitions[2n],?(MemberQ[ #,1]&)]]], {n,0,40}] (* _Harvey P. Dale, Aug 08 2013 *)
b[n_] := DivisorSigma[0, n]-1+Sum[(DivisorSigma[0, k]-1)*(PartitionsP[n-k] - PartitionsP[n-k-1]), {k, 1, n-1}]; a[0] = 0; a[n_] := b[2n]; Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Oct 07 2015 *)

More terms from Alois P. Heinz, Dec 03 2010

A182992 Number of parts of the n-th subshell of the head of the last section of the set of partitions of any even integer >= 2n.

Original entry on oeis.org

1, 2, 5, 9, 17, 34, 55, 96, 163, 260, 413, 658, 1000, 1528, 2304, 3405, 5003, 7302, 10497, 15022, 21330, 30010, 41980, 58395, 80607, 110740, 151334, 205623, 278119, 374513, 501842, 669828, 890461, 1178960, 1555314

Offset: 1

Omar E. Pol, Feb 06 2011

nonn

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

Also first differences of A182734. - Omar E. Pol, Mar 03 2011

			a(5)=17 because the 5th subshell of the head of the last section of any even integer >= 10 looks like this:
(10 . . . . . . . . . )
( 5 . . . . 5 . . . . )
( 6 . . . . . 4 . . . )
( 7 . . . . . . 3 . . )
( 4 . . . 3 . . 3 . . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
.                (2 . )
The subshell has 17 parts, so a(5)=17.

Cf. A135010, A138135, A182734, A182742, A182982, A182993, A182994.

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

More terms from Omar E. Pol, Mar 03 2011

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.

A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of the last section of the set of partitions of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 2, 2, 2, 1, 1, 4, 4, 3, 2, 1, 1, 4, 4, 4, 3, 2, 1, 1, 7, 7, 6, 5, 3, 2, 1, 1, 8, 8, 8, 6, 5, 3, 2, 1, 1, 12, 12, 11, 10, 7, 5, 3, 2, 1, 1, 14, 14, 14, 12, 10, 7, 5, 3, 2, 1, 1, 21, 21, 20, 18, 14, 11, 7, 5, 3, 2, 1, 1

Offset: 1

Omar E. Pol, Mar 10 2012

Note that for n >= 2 the tail of the last section of n starts at the second column and the second column contains only one part of size 1, thus both the first and the second columns contain the same number of parts. For more information see A135010 and A182703.

			Illustration of initial terms. First six rows of triangle as numbers of parts in the columns from the last sections of the first six natural numbers:
.                                       6
.                                       3 3
.                                       4 2
.                                       2 2 2
.                           5             1
.                           3 2             1
.                 4           1             1
.                 2 2           1             1
.         3         1           1             1
.   2       1         1           1             1
1     1       1         1           1             1
---------------------------------------------------
1,  1,1,  1,1,1,  2,2,1,1,  2,2,2,1,1,  4,4,3,2,1,1
...
Triangle begins:
1;
1,   1;
1,   1,  1;
2,   2,  1,  1;
2,   2,  2,  1,  1;
4,   4,  3,  2,  1,  1;
4,   4,  4,  3,  2,  1,  1;
7,   7,  6,  5,  3,  2,  1,  1;
8,   8,  8,  6,  5,  3,  2,  1,  1;
12, 12, 11, 10,  7,  5,  3,  2,  1,  1;
14, 14, 14, 12, 10,  7,  5,  3,  2,  1,  1;
21, 21, 20, 18, 14, 11,  7,  5,  3,  2,  1,  1;

Column 1 is A187219. Row sums give A138137. Reversed rows converge to A000041.

Cf. A002865, A058399, A135010, A138135, A181187, A207031, A207380.

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

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

A194796 Imbalance of the number of parts of all partitions of n.

Original entry on oeis.org

0, -1, 0, -3, 0, -8, 0, -17, 3, -31, 10, -58, 22, -101, 52, -167, 104, -278, 191, -451, 344, -711, 594, -1119, 983, -1730, 1606, -2635, 2555, -3990, 3978, -5972, 6118, -8835, 9269, -12986, 13835, -18917, 20454, -27320, 29900, -39204, 43268, -55846, 62112

Offset: 1

Omar E. Pol, Feb 01 2012

sign

Consider the three-dimensional structure of the shell model of partitions, version "tree" (see the illustration in A194795). 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. For more information see A135010.

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

Cf. A006128, A096541, A135010, A138121, A138135, A138137, A197595, A194797, A194809.

b:= proc(n, i) option remember; local f, g;
      if n=0 or i=1 then [1, 0]
    else f:= b(n, i-1); g:= `if`(i>n, [0, 0], b(n-i, i));
         [f[1]+g[1], f[2]+g[2]+g[1]]
      fi
    end:
a:= proc(n) option remember;
      (-1)^n*(b(n-1, n-1)[2]-b(n, n)[2])+`if`(n=1, 0, a(n-1))
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Apr 04 2012

b[n_, i_] := b[n, i] = Module[{f, g}, If[n == 0 || i == 1, {1, 0}, f = b[n, i-1]; g = If[i>n, {0, 0}, b[n-i, i]]; {f[[1]] + g[[1]], f[[2]] + g[[2]] + g[[1]]}]]; a[n_] := a[n] = (-1)^n*(b[n-1, n-1][[2]] - b[n, n][[2]]) + If[n == 1, 0, a[n-1]]; Table [a[n], {n, 1, 60}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

vector(50, n, sum(k=1, n, (-1)^(k-1)*(numdiv(k)-1+sum(j=1, k-1, (numdiv(j)-1)*(numbpart(k-j)-numbpart(k-j-1)))))) \\ Altug Alkan, Nov 11 2015

a(n) = Sum_{k=1..n} (-1)^(k-1)*A138135(k).

More terms from Alois P. Heinz, Apr 04 2012

A208474 Sum of the sizes of the Durfee squares of all partitions of n that do not contain 1 as a part, but with a(1) = 1.

Original entry on oeis.org

1, 1, 1, 3, 3, 7, 7, 13, 16, 24, 30, 46, 55, 79, 100, 136, 169, 229, 282, 374, 462, 598, 737, 947, 1158, 1466, 1794, 2246, 2733, 3399, 4116, 5076, 6133, 7503, 9033, 10993, 13177, 15943, 19061, 22939, 27327, 32749, 38883, 46395, 54938, 65278, 77070, 91270

Offset: 1

Omar E. Pol, Mar 03 2012

nonn

Also sum of the sizes of the Durfee squares of all partitions of the head of the last section of n (see A135010).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
G. E. Andrews, Partitions and Durfee Dissection
Eric Weisstein's World of Mathematics, Durfee Square

First differences of A115995.

Cf. A135010, A138135, A138137, A138879.

b:= proc(n, i) option remember; `if`(n=0, 1,
      `if`(i<1, 0, b(n, i-1)+`if`(i>n, 0, b(n-i, i))))
    end:
g:= proc(n) option remember;
      add(add(b(k, d)*b(n-d^2-k, d),
              k=0..n-d^2)*d, d=1..floor(sqrt(n)))
    end:
a:= n-> g(n)-g(n-1):
seq(a(n), n=1..70);  # Alois P. Heinz, Apr 09 2012

b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, b[n, i-1] + If[i>n, 0, b[n-i, i]]]]; g[n_] := Sum[Sum[b[k, d]*b[n-d^2-k, d], {k, 0, n-d^2}]*d, {d, 1, Sqrt[n]}]; Table[g[n], {n, 0, 70}] // Differences (* Jean-François Alcover, Feb 21 2017, after Alois P. Heinz *)

a(n) ~ log(2) * exp(Pi*sqrt(2*n/3)) / (4*n*sqrt(3)). - Vaclav Kotesovec, Jan 03 2019

cp's OEIS Frontend

A207035 Sum of all parts minus the total number of parts of the last section of the set of partitions of n.

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A182734 Number of parts in all partitions of 2n that do not contain 1 as a part.

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A182992 Number of parts of the n-th subshell of the head of the last section of the set of partitions of any even integer >= 2n.

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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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A207379 Triangle read by rows: T(n,k) = number of parts that are in the k-th column of 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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Maple

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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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Maple

Mathematica

Extensions

A194796 Imbalance of the number of parts of all partitions of n.

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