A235793 -id:A235793 - 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 *)

A235798 Triangle read by rows: T(n,k) = number of occurrences of k in all overpartitions of n.

Original entry on oeis.org

2, 4, 2, 10, 4, 2, 20, 8, 4, 2, 38, 16, 8, 4, 2, 68, 30, 16, 8, 4, 2, 118, 52, 28, 16, 8, 4, 2, 196, 88, 48, 28, 16, 8, 4, 2, 318, 144, 82, 48, 28, 16, 8, 4, 2, 504, 230, 132, 80, 48, 28, 16, 8, 4, 2, 782, 360, 208, 128, 80, 48, 28, 16, 8, 4, 2, 1192, 552, 324, 202, 128, 80, 48, 28, 16, 8, 4, 2

Offset: 1

Omar E. Pol, Jan 18 2014

It appears that row n lists the first differences of row n of triangle A235797 together with 2 (as the final term of the row).

The equivalent sequence for partitions is A066633.

			Triangle begins:
2;
4,   2;
10,  4,  2;
20,  8,  4,  2;
38, 16,  8,  4,  2;
68, 30, 16,  8,  4,  2;
...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)

Row sums give A235792.

Cf. A015128, A066633, A235790, A235793, A235797, A236000, A236001.

A(n)={my(p=prod(k=1, n, (1 + x^k)/(1 - x^k) + O(x*x^n))); Mat(vector(n, k, Col(2*(p + O(x*x^(n-k)))*x^k/((1 - x^k)*(1 + x^k)), -n)))}
{ my(T=A(10)); for(n=1, #T, print(T[n, 1..n])) } \\ Andrew Howroyd, Feb 19 2020

G.f. of column k: 2*(x^k/((1 - x^k)*(1 + x^k))) * Product_{j>0} (1 + x^j)/(1 - x^j). - Andrew Howroyd, Feb 19 2020

Terms a(22) and beyond from Andrew Howroyd, Feb 19 2020

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

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

A236626 Sum of all parts of all overcompositions of n.

Original entry on oeis.org

2, 8, 36, 104, 300, 864, 2268, 5824, 14418, 35760, 85888, 204816, 479804, 1113280, 2560560, 5836704, 13209612, 29690208, 66332572, 147350880, 325780056, 716862256, 1571067072, 3429697920, 7461222850, 16178111560, 34973640108, 75392349648

Offset: 1

Omar E. Pol, Feb 01 2014

nonn

For the definition of overcomposition see A236002.

The equivalent sequence for overpartitions is A235793.

			For n = 3 the 12 overcompositions of 3 are [3], [3'], [1, 2], [1', 2], [1, 2'], [1', 2'], [2, 1], [2', 1], [2, 1'], [2', 1'], [1, 1, 1], [1', 1, 1], hence the sum of all parts is 3+3+1+2+1+2+1+2+1+2+2+1+2+1+2+1+2+1+1+1+1+1+1+1 = 3*12 = 36, so a(3) = 36.

Cf. A001787, A066186, A235793, A235999, A236002, A236625, A236628, A236633.

a(n) = n*A236002(n).

cp's OEIS Frontend

A235792 Total number of parts in all overpartitions of n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A235798 Triangle read by rows: T(n,k) = number of occurrences of k in all overpartitions of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Comments

Examples

Crossrefs

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

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

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

PARI

Formula

A236626 Sum of all parts of all overcompositions of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula