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Total number of parts in all overpartitions of n.
+10
15
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
COMMENTS
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.
MAPLE
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]:
MATHEMATICA
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 *)
Triangle read by rows: T(n,k) = 2^k* A116608(n,k), n>=1, k>=1.
+10
13
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
COMMENTS
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).
EXAMPLE
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;
...
MAPLE
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)):
MATHEMATICA
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 *)
CROSSREFS
Cf. A000217, A003056, A116608, A196020, A211971, A235792, A235793, A235797, A235798, A235999, A236000, A236001.
Sum of all parts of all overpartitions of n.
+10
9
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
COMMENTS
The equivalent sequence for partitions is A066186.
MAPLE
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]:
Triangle read by rows in which row n lists the overpartitions of n in colexicographic order.
+10
8
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
COMMENTS
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.
EXAMPLE
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;
.
Sum of positive ranks of all overpartitions of n.
+10
7
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
COMMENTS
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.
EXAMPLE
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.
PROG
(PARI) 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
Triangle read by rows in which T(n,k) is the sum of the k-th largest elements in all overpartitions of n.
+10
6
2, 6, 2, 16, 6, 2, 34, 14, 6, 2, 68, 30, 14, 6, 2, 128, 60, 30, 14, 6, 2
COMMENTS
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.
EXAMPLE
Triangle begins:
2;
6, 2;
16, 6, 2;
34, 14, 6, 2;
68, 30, 14, 6, 2;
128, 60, 30, 14, 6, 2;
...
a(n) is the sum, over all overpartitions of n, of the non-overlined parts.
+10
3
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
FORMULA
G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) * Sum_{n>=1} n*q^n/(1-q^n).
EXAMPLE
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.
PROG
(PARI) 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
CROSSREFS
Cf. A305102 (number of non-overlined parts).
a(n) is the sum, over all overpartitions of n, of the overlined parts.
+10
3
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
FORMULA
G.f.: (Product_{k>=1} (1+q^k)/(1-q^k)) * Sum_{n>=1} n*q^n/(1+q^n).
EXAMPLE
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.
PROG
(PARI) 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
CROSSREFS
Cf. A305101 (number of overlined parts).
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