A235798 -id:A235798 - cp's OEIS frontend

A235792 Total number of parts in all overpartitions of n.

2, 6, 16, 34, 68, 128, 228, 390, 650, 1052, 1664, 2584, 3940, 5916, 8768, 12826, 18552, 26566, 37672, 52956, 73848, 102192, 140420, 191688, 260038, 350700, 470384, 627604, 833236, 1101080, 1448500, 1897438, 2475464, 3217016, 4165200, 5373714, 6909180, 8854288

Offset: 1

Omar E. Pol, Jan 18 2014

nonn

It appears that a(n) is also the sum of largest parts of all overpartitions of n.
More generally, It appears that the total number of parts >= k in all overpartitions of n equals the sum of k-th largest parts of all overpartitions of n. In this case k = 1. Also the first column of A235797.
The equivalent sequence for partitions is A006128.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Alois P. Heinz)

Cf. A006128, A015128, A211971, A235790, A235793, A235797, A235798, A236000, A236001.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0$2], b(n, i-1)+add((l-> l+[0, l[1]*j])
       (2*b(n-i*j, i-1)), j=1..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 21 2014

b[n_, i_] := b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, b[n, i-1] + Sum[ Function[l, l+{0, l[[1]]*j}][2*b[n-i*j, i-1]], {j, 1, n/i}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Nov 11 2015, after Alois P. Heinz *)

log(a(n)) ~ Pi*sqrt(n). - Vaclav Kotesovec, Apr 13 2025

More terms from Alois P. Heinz, Jan 21 2014

A235790 Triangle read by rows: T(n,k) = 2^k*A116608(n,k), n>=1, k>=1.

Original entry on oeis.org

2, 4, 4, 4, 6, 8, 4, 20, 8, 24, 8, 4, 44, 16, 8, 52, 40, 6, 68, 80, 8, 88, 120, 16, 4, 108, 200, 32, 12, 116, 296, 80, 4, 148, 416, 160, 8, 176, 536, 320, 8, 176, 776, 480, 32, 10, 220, 936, 832, 64, 4, 236, 1232, 1232, 160, 12, 272, 1472, 1872, 320

Offset: 1

Omar E. Pol, Jan 18 2014

It appears that T(n,k) is the number of overpartitions of n having k distinct parts. (This is true by definition, Joerg Arndt, Jan 20 2014).
Row n has length A003056(n) hence the first element of column k is in row A000217(k).
The first element of column k is A000079(k).

			Triangle begins:
2;
4;
4,    4;
6,    8;
4,   20;
8,   24,    8;
4,   44,   16;
8,   52,   40;
6,   68,   80;
8,   88,  120,   16;
4,  108,  200,   32;
12, 116,  296,   80;
4,  148,  416,  160;
8,  176,  536,  320;
8,  176,  776,  480,   32;
10, 220,  936,  832,   64;
4,  236, 1232, 1232,  160;
12, 272, 1472, 1872,  320;
4,  284, 1880, 2592,  640;
12, 324, 2216, 3632, 1152;
8,  328, 2704, 4944, 1856, 64;
...

Alois P. Heinz, Rows n = 1..500, flattened

Row sums give A015128, n >= 1.
Column 1 is A062011.
Cf. A000217, A003056, A116608, A196020, A211971, A235792, A235793, A235797, A235798, A235999, A236000, A236001.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      expand(b(n, i-1)+add(x*b(n-i*j, i-1), j=1..n/i))))
    end:
T:= n->(p->seq(2^i*coeff(p, x, i), i=1..degree(p)))(b(n$2)):
seq(T(n), n=1..20);  # Alois P. Heinz, Jan 20 2014

b[n_, i_] := b[n, i] = If[n == 0, 1, If[i<1, 0, Expand[b[n, i-1] + Sum[x*b[n-i*j, i-1], {j, 1, n/i}]]]]; T[n_] := Function[p, Table[2^i * Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][b[n, n]]; Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, Oct 20 2016, after Alois P. Heinz *)

A235793 Sum of all parts of all overpartitions of n.

Original entry on oeis.org

2, 8, 24, 56, 120, 240, 448, 800, 1386, 2320, 3784, 6048, 9464, 14560, 22080, 32992, 48688, 71064, 102600, 146720, 207984, 292336, 407744, 564672, 776650, 1061424, 1442016, 1947904, 2617192, 3498720, 4654464, 6163584, 8126448, 10669472, 13952400, 18175896

Offset: 1

Omar E. Pol, Jan 18 2014

nonn

The equivalent sequence for partitions is A066186.

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

Cf. A015128, A066186, A235790, A235792, A235797, A235798, A236000.

b:= proc(n, i) option remember; `if`(n=0, [1, 0],
      `if`(i<1, [0$2], b(n, i-1)+add((l-> l+[0, l[1]*i*j])
       (2*b(n-i*j, i-1)), j=1..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..40);  # Alois P. Heinz, Jan 21 2014

Table[n*Sum[PartitionsP[n-k]*PartitionsQ[k], {k, 0, n}], {n, 1, 40}] (* Jean-François Alcover, Oct 20 2016, after Vaclav Kotesovec *)

a(n) = n*A015128(n).

a(n) ~ exp(Pi*sqrt(n)) / 8. - Vaclav Kotesovec, May 19 2018

A236000 Triangle read by rows in which row n lists the overpartitions of n in colexicographic order.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 3, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 3, 1, 3, 1, 3, 1, 3, 1, 2, 2, 2, 2, 4, 4, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1

Offset: 1

Omar E. Pol, Jan 18 2014

In the data section the overlined parts cannot be represented correctly, therefore the sequence represents all possible suborderings generated by the overlined parts.
The diagram in the second part of the Example section shows only one of the possible suborderings.
The equivalent sequence for partitions is A211992.
The equivalent sequence for compositions is A228525.
See both sequences for more information.
Row n contains A015128(n) overpartitions.
Row n contains A235792(n) parts.
Row sums give A235793.

			Triangle begins:
[1], [1];
[1, 1], [1, 1], [2], [2];
[1, 1, 1], [1, 1, 1], [2, 1], [2, 1], [2, 1], [2, 1], [3], [3];
[1, 1, 1, 1], [1, 1, 1, 1], [2, 1, 1], [2, 1, 1], [2, 1, 1], [2, 1, 1], [3, 1], [3, 1], [3, 1], [3, 1], [2, 2], [2, 2], [4], [4];
...
Illustration of initial terms (n: 1..4)
-----------------------------------------
n      Diagram          Overpartition
-----------------------------------------
.       _
1      |.|              1',
1      |_|              1;
.       _ _
2      |.| |            1', 1,
2      |_| |            1,  1,
2      |  .|            2',
2      |_ _|            2;
.       _ _ _
3      |.| | |          1', 1,  1,
3      |_| | |          1,  1,  1,
3      |  .|.|          2', 1',
3      |   |.|          2,  1',
3      |  .| |          2', 1,
3      |_ _| |          2,  1,
3      |    .|          3',
3      |_ _ _|          3;
.       _ _ _ _
4      |.| | | |        1', 1,  1,  1,
4      |_| | | |        1,  1,  1,  1,
4      |  .|.| |        2', 1', 1,
4      |   |.| |        2,  1', 1,
4      |  .| | |        2', 1,  1,
4      |_ _| | |        2,  1,  1,
4      |    .|.|        3', 1',
4      |     |.|        3,  1',
4      |    .| |        3', 1,
4      |_ _ _| |        3,  1,
4      |  .|   |        2', 2,
4      |_ _|   |        2,  2,
4      |      .|        4',
4      |_ _ _ _|        4;
.

Cf. A015128, A211992, A228525, A235790, A235792, A235793, A235797, A235798, A236001

A236001 Sum of positive ranks of all overpartitions of n.

Original entry on oeis.org

0, 2, 4, 10, 20, 36, 64, 110, 180, 288, 452, 696, 1052, 1568, 2304, 3346, 4808, 6838, 9636, 13464, 18664, 25684, 35104, 47672, 64348, 86368, 115304, 153152, 202452, 266404, 349032, 455406, 591856, 766284, 988544, 1270862, 1628380, 2079828, 2648296, 3362180

Offset: 1

Omar E. Pol, Jan 18 2014

nonn

Consider here that the rank of a overpartition is the largest part minus the number of parts (the same idea as the Dyson's rank of a partition).
It appears that the sum of all ranks of all overpartitions of n is equal to zero.
The equivalent sequence for partitions is A209616.

			For n = 4 we have:
---------------------------
Overpartitions
of 4               Rank
---------------------------
4               4 - 1 =  3
4               4 - 1 =  3
2+2             2 - 2 =  0
2+2             2 - 2 =  0
3+1             3 - 2 =  1
3+1             3 - 2 =  1
3+1             3 - 2 =  1
3+1             3 - 2 =  1
2+1+1           2 - 3 = -1
2+1+1           2 - 3 = -1
2+1+1           2 - 3 = -1
2+1+1           2 - 3 = -1
1+1+1+1         1 - 4 = -3
1+1+1+1         1 - 4 = -3
---------------------------
The sum of positive ranks of all overpartitions of 4 is 3+3+1+1+1+1 = 10 so a(4) = 10.

Cf. A015128, A209616, A235790, A235792, A235797, A235798, A236000.

a(n)={my(s=0); forpart(p=n, my(r=p[#p]-#p); if(r>0, s+=r*2^#Set(p))); s} \\ Andrew Howroyd, Feb 19 2020

Terms a(7) and beyond from Andrew Howroyd, Feb 19 2020

A235797 Triangle read by rows in which T(n,k) is the sum of the k-th largest elements in all overpartitions of n.

Original entry on oeis.org

2, 6, 2, 16, 6, 2, 34, 14, 6, 2, 68, 30, 14, 6, 2, 128, 60, 30, 14, 6, 2

Offset: 1

Omar E. Pol, Jan 18 2014

It appears that T(n,k) is also the total number of parts >= k in all overpartitions of n.
It appears that the first differences of row n together with 2 give row n of triangle A235798.
The equivalent sequence for partitions is A181187.

			Triangle begins:
    2;
    6,  2;
   16,  6,  2;
   34, 14,  6,  2;
   68, 30, 14,  6,  2;
  128, 60, 30, 14,  6,  2;
  ...

Column 1 is A235792.

Cf. A015128, A181187, A235790, A235793, A235798, A236000, A236001.

A335651 a(n) is the sum, over all overpartitions of n, of the non-overlined parts.

Original entry on oeis.org

1, 5, 14, 35, 74, 150, 280, 505, 875, 1470, 2402, 3850, 6034, 9300, 14120, 21131, 31220, 45619, 65930, 94374, 133892, 188350, 262904, 364350, 501459, 685762, 932200, 1259944, 1693750, 2265380, 3015152, 3994585, 5268988, 6920700, 9053748, 11798873, 15319610, 19820738, 25557560

Offset: 1

Jeremy Lovejoy, Jun 17 2020

nonn

			The 8 overpartitions of 3 are [3], [3'], [2,1], [2,1'], [2',1], [2',1'], [1,1,1], [1',1,1], and so a(3) = 14.

K. Bringmann, J. Lovejoy, and R. Osburn, Rank and crank moments for overpartitions, Journal of Number Theory, 129 (2009), 1758-1772.

Cf. A015128, A230441, A235790, A235792, A235793, A235798, A236000, A236001, A237047, A335666.

Cf. A305102 (number of non-overlined parts).

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, k*q^k/(1-q^k)) ) \\ Joerg Arndt, Jun 18 2020

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) * Sum_{n>=1} n*q^n/(1-q^n).

a(n) = A235793(n) - A335666(n). - Omar E. Pol, Jun 17 2020

A335666 a(n) is the sum, over all overpartitions of n, of the overlined parts.

Original entry on oeis.org

1, 3, 10, 21, 46, 90, 168, 295, 511, 850, 1382, 2198, 3430, 5260, 7960, 11861, 17468, 25445, 36670, 52346, 74092, 103986, 144840, 200322, 275191, 375662, 509816, 687960, 923442, 1233340, 1639312, 2168999, 2857460, 3748772, 4898652, 6377023, 8271294, 10690830, 13771912, 17683642

Offset: 1

Jeremy Lovejoy, Jun 17 2020

nonn

			The 8 overpartitions of 8 are [3], [3'], [2,1], [2,1'], [2',1], [2',1'], [1,1,1], [1',1,1], and so a(3) = 10.

K. Bringmann, J. Lovejoy, and R. Osburn, Rank and crank moments for overpartitions, Journal of Number Theory, 129 (2009), 1758-1772.

Cf. A015128, A230441, A235790, A235792, A235793, A235798, A236000, A236001, A237047, A335651.

Cf. A305101 (number of overlined parts).

my(N=44, q='q+O('q^N)); Vec( prod(k=1,N, (1+q^k)/(1-q^k)) * sum(k=1,N, k*q^k/(1+q^k)) ) \\ Joerg Arndt, Jun 18 2020

G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) * Sum_{n>=1} n*q^n/(1+q^n).

a(n) = A235793(n) - A335651(n). - Omar E. Pol, Jun 17 2020

cp's OEIS Frontend

A235792 Total number of parts in all overpartitions of n.

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A235790 Triangle read by rows: T(n,k) = 2^k*A116608(n,k), n>=1, k>=1.

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A235793 Sum of all parts of all overpartitions of n.

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A236000 Triangle read by rows in which row n lists the overpartitions of n in colexicographic order.

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A236001 Sum of positive ranks of all overpartitions of n.

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A235797 Triangle read by rows in which T(n,k) is the sum of the k-th largest elements in all overpartitions of n.

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A335651 a(n) is the sum, over all overpartitions of n, of the non-overlined parts.

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A335666 a(n) is the sum, over all overpartitions of n, of the overlined parts.

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