A032302 -id:A032302 - cp's OEIS frontend

A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 6, 2, 6, 4, 4, 2, 6, 3, 4, 4, 6, 2, 7, 2, 6, 4, 4, 4, 7, 2, 4, 4, 7, 2, 8, 2, 6, 6, 4, 2, 7, 3, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 8, 2, 4, 5, 7, 4, 8, 2, 6, 4, 7, 2, 8, 2, 4, 6, 6, 4, 8, 2, 8, 5, 4, 2, 9, 4, 4, 4

Offset: 1

Gus Wiseman, Feb 17 2018

nonn

An integer n is a subset-sum of an integer partition y if there exists a submultiset of y with sum n. The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

Position of first appearance of n appears to be A259941(n-1) = least Heinz number of a complete partition of n-1. - Gus Wiseman, Nov 16 2023

			The subset-sums of (5,1,1,1) are {0, 1, 2, 3, 5, 6, 7, 8} so a(88) = 8.
The subset-sums of (4,3,1) are {0, 1, 3, 4, 5, 7, 8} so a(70) = 7.

Alois P. Heinz, Table of n, a(n) for n = 1..20000

Cf. A000005, A000041, A000720, A001222, A056239, A108917, A112798, A122111, A122768, A215366, A276024, A284640, A296150, A299702.
Positions of first appearances are A259941.
The triangle for this rank statistic is A365658.
The semi version is A366739, sum A366738, strict A366741.
Cf. A032302, A070933, A304792, A304793, A365543, A365920, A367095.

Table[Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]],{n,100}]

a(n) <= A000005(n) and a(n) = A000005(n) iff n is the Heinz number of a knapsack partition (A299702).

Comment corrected by Gus Wiseman, Aug 09 2024

A072706 Number of unimodal partitions/compositions of n into distinct terms.

Original entry on oeis.org

1, 1, 1, 3, 3, 5, 9, 11, 15, 21, 33, 39, 55, 69, 93, 127, 159, 201, 261, 327, 411, 537, 653, 819, 1011, 1257, 1529, 1899, 2331, 2829, 3441, 4179, 5031, 6093, 7305, 8767, 10575, 12573, 14997, 17847, 21223, 25089, 29757, 35055, 41379, 48801, 57285, 67131

Offset: 0

Henry Bottomley, Jul 04 2002

nonn

Also the number of ways to partition a strict integer partition of n into two unordered blocks. - Gus Wiseman, Dec 31 2019

			a(6)=9 since 6 can be written as 1+2+3, 1+3+2, 1+5, 2+3+1, 2+4, 3+2+1, 4+2, 5+1, or 6, but not for example 1+4+1 (which does not have distinct terms) nor 2+1+3 (which is not unimodal).
From _Joerg Arndt_, Mar 25 2014: (Start)
The a(10) = 33 such compositions of 10 are:
01:  [ 1 2 3 4 ]
02:  [ 1 2 4 3 ]
03:  [ 1 2 7 ]
04:  [ 1 3 4 2 ]
05:  [ 1 3 6 ]
06:  [ 1 4 3 2 ]
07:  [ 1 4 5 ]
08:  [ 1 5 4 ]
09:  [ 1 6 3 ]
10:  [ 1 7 2 ]
11:  [ 1 9 ]
12:  [ 2 3 4 1 ]
13:  [ 2 3 5 ]
14:  [ 2 4 3 1 ]
15:  [ 2 5 3 ]
16:  [ 2 7 1 ]
17:  [ 2 8 ]
18:  [ 3 4 2 1 ]
19:  [ 3 5 2 ]
20:  [ 3 6 1 ]
21:  [ 3 7 ]
22:  [ 4 3 2 1 ]
23:  [ 4 5 1 ]
24:  [ 4 6 ]
25:  [ 5 3 2 ]
26:  [ 5 4 1 ]
27:  [ 6 3 1 ]
28:  [ 6 4 ]
29:  [ 7 2 1 ]
30:  [ 7 3 ]
31:  [ 8 2 ]
32:  [ 9 1 ]
33:  [ 10 ]
(End)

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

The non-strict version is A001523.

Cf. A000009, A000041, A001970, A032020, A059618, A072705, A072707, A270995, A294617.

b:= proc(n, i) option remember; `if`(n>i*(i+1)/2, 0, `if`(n=0, 1,
      expand(b(n, i-1) +`if`(i>n, 0, x*b(n-i, i-1)))))
    end:
a:= n->(p->add(coeff(p, x, i)*ceil(2^(i-1)), i=0..degree(p)))(b(n$2)):
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 25 2014

b[n_, i_] := b[n, i] = If[n > i*(i + 1)/2, 0, If[n == 0, 1, Expand[b[n, i - 1] + If[i > n, 0, x*b[n - i, i - 1]]]]]; a[n_] := Function[{p}, Sum[Coefficient[p, x, i]*Ceiling[2^(i - 1)], {i, 0, Exponent[p, x]}]][b[n, n]]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jan 16 2015, after Alois P. Heinz *)
Table[If[n==0,1,Sum[2^(Length[ptn]-1),{ptn,Select[IntegerPartitions[n],UnsameQ@@#&]}]],{n,0,15}] (* Gus Wiseman, Dec 31 2019 *)

N=66; q='q+O('q^N); Vec( 1 + sum(n=1, N, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1, n, 1-q^k ) ) ) \\ Joerg Arndt, Mar 25 2014

a(n) = sum_k A072705(n, k) = A032020(n)-A072707(k) = A032302(n)/2 (n>0).
G.f.: 1/2*(1+Product_{k>0} (1+2*x^k)). - Vladeta Jovovic, Jun 24 2003
G.f.: 1 + sum(n>=1, 2^(n-1)*q^(n*(n+1)/2) / prod(k=1..n, 1-q^k ) ). [Joerg Arndt, Jan 20 2014]
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(3*Pi)*n^(3/4)), where c = -polylog(2, -2) = A266576 = 1.436746366883680946362902023893583354... - Vaclav Kotesovec, Sep 22 2019

A237258 Number of strict partitions of 2n that include a partition of n.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 32, 43, 63, 84, 122, 158, 220, 293, 393, 511, 685, 881, 1156, 1485, 1925, 2445, 3147, 3952, 5019, 6323, 7924, 9862, 12336, 15259, 18900, 23294, 28646, 35091, 42985, 52341, 63694, 77336, 93588, 112973, 136367, 163874, 196638

Offset: 0

Clark Kimberling, Feb 05 2014

nonn

A strict partition is a partition into distinct parts.

			a(5) counts these partitions of 10: [5,4,1], [5,3,2], [4,3,2,1].

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..200

Cf. A000009, A237194, A235130.
The non-strict version is A002219, ranked by A357976.
These partitions are ranked by A357854.
A000712 counts distinct submultisets of partitions, strict A032302.
A304792 counts subset-sums of partitions, positive A276024, strict A284640.
Cf. A006827, A064914, A108917, A276107, A300061, A357879.

z = 24; Table[theTotals = Map[{#, Map[Total, Subsets[#]]} &,  Select[IntegerPartitions[2 nn], # == DeleteDuplicates[#] &]]; Length[Map[#[[1]] &, Select[theTotals, Length[Position[#[[2]], nn]] >= 1 &]]], {nn, z}] (* Peter J. C. Moses, Feb 04 2014 *)

a(n) = A237194(2n,n).

a(31)-a(47) from Alois P. Heinz, Feb 07 2014

A032308 Expansion of Product_{k>=1} (1 + 3*x^k).

Original entry on oeis.org

1, 3, 3, 12, 12, 21, 48, 57, 84, 120, 228, 264, 399, 516, 732, 1119, 1416, 1884, 2532, 3324, 4296, 6168, 7545, 9984, 12684, 16500, 20577, 26688, 34572, 43032, 54264, 68232, 84972, 106176, 131664, 162507, 205680, 249888, 308856, 377796, 465195, 564024, 691788, 835572, 1017768, 1241040

Offset: 0

Christian G. Bower

nonn

"EFK" (unordered, size, unlabeled) transform of 3,3,3,3,...

Number of partitions into distinct parts of 3 sorts, see example. [Joerg Arndt, May 22 2013]

			From _Joerg Arndt_, May 22 2013: (Start)
There are a(5) = 21 partitions of 5 into distinct parts of 3 sorts (format P:S for part:sort):
01:  [ 1:0  4:0  ]
02:  [ 1:0  4:1  ]
03:  [ 1:0  4:2  ]
04:  [ 1:1  4:0  ]
05:  [ 1:1  4:1  ]
06:  [ 1:1  4:2  ]
07:  [ 1:2  4:0  ]
08:  [ 1:2  4:1  ]
09:  [ 1:2  4:2  ]
10:  [ 2:0  3:0  ]
11:  [ 2:0  3:1  ]
12:  [ 2:0  3:2  ]
13:  [ 2:1  3:0  ]
14:  [ 2:1  3:1  ]
15:  [ 2:1  3:2  ]
16:  [ 2:2  3:0  ]
17:  [ 2:2  3:1  ]
18:  [ 2:2  3:2  ]
19:  [ 5:0  ]
20:  [ 5:1  ]
21:  [ 5:2  ]
(End)

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
C. G. Bower, Transforms (2)

Cf. A000009, A032302, A261568, A261569.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 3*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015
# Alternatively:
simplify(expand(QDifferenceEquations:-QPochhammer(-3,x,99),x)/4):
seq(coeff(%,x,n), n=0..45); # Peter Luschny, Nov 17 2016

nmax = 40; CoefficientList[Series[Product[1 + 3*x^k, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 24 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*3^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-3, x]/4 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

N=66; x='x+O('x^N); Vec(prod(n=1,N, 1+3*x^n)) \\ Joerg Arndt, May 22 2013

G.f.: Product_{k>=1} (1 + 3*x^k).
a(n) = (1/4) * [x^n] QPochammer(-3, x). - Vladimir Reshetnikov, Nov 20 2015
a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (4*sqrt(Pi)*n^(3/4)), where c = Pi^2/6 + log(3)^2/2 + polylog(2, -1/3) = 1.93937542076670895307727171917789144122... . - Vaclav Kotesovec, Jan 04 2016
G.f.: Sum_{i>=0} 3^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

a(0) prepended and more terms added by Joerg Arndt, May 22 2013

A266891 Expansion of Product_{k>=1} (1 + k*x^k)^k.

Original entry on oeis.org

1, 1, 4, 13, 29, 81, 188, 456, 1030, 2405, 5295, 11611, 25246, 53552, 113332, 235685, 486011, 990840, 2006567, 4018010, 7992003, 15768511, 30875424, 60060509, 116042548, 222817961, 425200270, 806991037, 1522748592, 2858792520, 5339457208, 9924370365

Offset: 0

Vaclav Kotesovec, Jan 05 2016

nonn

This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -n, g(n) = -n. - Seiichi Manyama, Nov 18 2017

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..5000 from Vaclav Kotesovec)
Vaclav Kotesovec, Graph - The asymptotic ratio (200000 terms)

Cf. A022629, A026007, A032302, A261562, A266964, A304210, A304211.

nmax=50; CoefficientList[Series[Product[(1+k*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
(* More efficient program: *) nmax = 50; s = 1+x; Do[s*=Sum[Binomial[k, j] * k^j * x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Jan 07 2016 *)

a(0) = 1 and a(n) = (1/n) * Sum_{k=1..n} b(k)*a(n-k) where b(n) = Sum_{d|n} d*(-d)^(1+n/d). - Seiichi Manyama, Nov 18 2017

Conjecture: log(a(n)) ~ n^(2/3) * (2*log(3*n) - 3) / (4*3^(1/3)). - Vaclav Kotesovec, May 08 2018

A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.

Original entry on oeis.org

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 5, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 8, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 7, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 8, 3, 3, 3, 5, 1, 7, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5, 1, 7, 1, 5, 5

Offset: 1

Gus Wiseman, Jun 29 2018

nonn

A rational number q is a nonempty-subset-average of an integer partition y if there exists a nonempty submultiset of y with average q.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k).

			The a(42) = 7 subset-averages of (4,2,1) are 1, 3/2, 2, 7/3, 5/2, 3, 4.
The a(72) = 7 subset-averages of (2,2,1,1,1) are 1, 5/4, 4/3, 7/5, 3/2, 5/3, 2.

Cf. A032302, A056239, A108917, A122768, A275972, A276024, A296150, A299701, A299702, A301899, A301957, A304793, A316313.

One less than A316398.

primeMS[n_]:=If[n===1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Table[Length[Union[Mean/@Rest[Subsets[primeMS[n]]]]],{n,100}]

up_to = 65537;
A056239(n) = { my(f); if(1==n, 0, f=factor(n); sum(i=1, #f~, f[i,2] * primepi(f[i,1]))); }
v056239 = vector(up_to,n,A056239(n));
A316314(n) = { my(m=Map(),s,k=0); fordiv(n,d,if((d>1)&&!mapisdefined(m,s = v056239[d]/bigomega(d)), mapput(m,s,s); k++)); (k); }; \\ Antti Karttunen, Sep 23 2018

a(n) = A316398(n) - 1.

More terms from Antti Karttunen, Sep 23 2018

A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

Original entry on oeis.org

1, 0, 0, 1, 1, 3, 4, 7, 9, 16, 21, 33, 46, 68, 95, 140, 187, 266, 372, 507, 683, 948, 1256, 1692, 2263, 3003, 3955, 5248, 6824, 8921, 11669, 15058, 19413, 25128, 32149, 41129, 52578, 66740, 84696, 107389, 135310, 170277, 214386, 268151, 335261, 418896, 521204

Offset: 0

Wouter Meeussen, Jul 09 2005

nonn

Counted as orderless pairs since intersection is commutative.

			Of the partitions of 12 into different parts, the partition (5+4+2+1) has an empty intersection with only (12) and (9+3).
From _Gus Wiseman_, Oct 07 2023: (Start)
The a(6) = 4 pairs are:
  ((6),(5,1))
  ((6),(4,2))
  ((6),(3,2,1))
  ((5,1),(4,2))
(End)

Vaclav Kotesovec Table of n, a(n) for n = 0..1050 (terms 0..700 from Alois P. Heinz)

Column k=2 of A258280.
Cf. A079122, A086737.
Main diagonal of A284593 times (1/2).
This is the strict case of A260669.
The ordered version is A365662 = strict case of A054440.
This is the disjoint case of A366132, with twins A366317.
A000041 counts integer partitions, strict A000009.
A002219 counts biquanimous partitions, strict A237258, ordered A064914.
Cf. A000124, A000712, A001255, A006827, A032302, A122768, A355389.

using DiscreteMath`Combinatorica`and ListPartitionsQ[n_Integer]:= Flatten[ Reverse /@ Table[(Range[m-1, 0, -1]+#1&)/@ TransposePartition/@ Complement[Partitions[ n-m* (m-1)/2, m], Partitions[n-m*(m-1)/2, m-1]], {m, -1+Floor[1/2*(1+Sqrt[1+8*n])]}], 1]; Table[Plus@@Flatten[Outer[If[Intersection[Flatten[ #1], Flatten[ #2]]==={}, 1, 0]&, ListPartitionsQ[k], ListPartitionsQ[k], 1]], {k, 48}]/2
nmax = 50; p = 1; Do[p = Expand[p*(1 + x^j + y^j)]; p = Select[p, (Exponent[#, x] <= nmax) && (Exponent[#, y] <= nmax) &], {j, 1, nmax}]; p = Select[p, Exponent[#, x] == Exponent[#, y] &]; Table[Coefficient[p, x^n*y^n]/2, {n, 1, nmax}] (* Vaclav Kotesovec, Apr 07 2017 *)
Table[Length[Select[Subsets[Select[IntegerPartitions[n], UnsameQ@@#&],{2}],Intersection@@#=={}&]],{n,15}] (* Gus Wiseman, Oct 07 2023 *)

a(n) = {my(A=1 + O(x*x^n) + O(y*y^n)); polcoef(polcoef(prod(k=1, n, A + x^k + y^k), n), n)/2} \\ Andrew Howroyd, Oct 10 2023

a(n) = ceiling(1/2 * [(x*y)^n] Product_{j>0} (1+x^j+y^j)). - Alois P. Heinz, Mar 31 2017

a(n) = ceiling(A365662(n)/2). - Gus Wiseman, Oct 07 2023

Name edited by Gus Wiseman, Oct 10 2023

a(0)=1 prepended by Alois P. Heinz, Feb 09 2024

A261584 Expansion of Product_{k>=1} (1 + 2x^k)/(1 - 2x^k).

Original entry on oeis.org

1, 4, 12, 36, 92, 228, 540, 1236, 2748, 6004, 12876, 27252, 57036, 118308, 243564, 498564, 1015484, 2060484, 4167804, 8409588, 16934748, 34049940, 68378220, 137185428, 275026476, 551052676, 1103618508, 2209525092, 4422484764, 8850120420, 17707920924

Offset: 0

Vaclav Kotesovec, Aug 25 2015

nonn

Cf. A032302, A070933, A261563.

nmax = 40; CoefficientList[Series[Product[(1 + 2*x^k)/(1 - 2*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[2^(2*k)/(2*k-1)*x^(2*k-1)/(1 - x^(2*k-1)), {k, 1, nmax}]], {x, 0, nmax}], x]
(O[x]^30 - QPochhammer[-2, x]/(3 QPochhammer[2, x]))[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) = c * 2^n, where c = 1/(A048651 * A083864) = 2*Product_{j>=1} (2^j+1)/(2^j-1) = 16.5119758715565001310882816988645462530540032335764606912075051272567456...

A261568 Expansion of Product_{k>=1} (1 + 4*x^k).

Original entry on oeis.org

1, 4, 4, 20, 20, 36, 100, 116, 180, 260, 580, 660, 1044, 1380, 2020, 3444, 4340, 6020, 8260, 11220, 14740, 23140, 28196, 38900, 50420, 67780, 85956, 114900, 157140, 197860, 257060, 331060, 423540, 540100, 687620, 864084, 1145300, 1406500, 1789860, 2231860

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

Cf. A000009, A032302, A032308, A261569.

b:= proc(n, i) option remember; `if`(i*(i+1)/2n, 0, 4*b(n-i, i-1))))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 24 2015

nmax = 40; CoefficientList[Series[Product[1 + 4*x^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 40; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*4^k/k*x^k/(1-x^k), {k, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 25 2015 *)
(QPochhammer[-4, x]/5 + O[x]^58)[[3]] (* Vladimir Reshetnikov, Nov 20 2015 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(5*Pi)*n^(3/4)), where c = Pi^2/6 + 2*log(2)^2 + polylog(2, -1/4) = 2.36993979699836583198553742535032304875... . - Vaclav Kotesovec, Jan 04 2016

G.f.: Sum_{i>=0} 4^i*x^(i*(i+1)/2)/Product_{j=1..i} (1 - x^j). - Ilya Gutkovskiy, Apr 12 2018

A261562 Expansion of Product_{k>=1} (1 + 2*x^k)^k.

Original entry on oeis.org

1, 2, 4, 14, 24, 58, 124, 238, 480, 922, 1764, 3238, 6008, 10794, 19292, 34166, 59504, 103042, 176452, 299958, 505240, 845570, 1403324, 2315118, 3794640, 6180370, 10009540, 16121374, 25829512, 41171690, 65320956, 103140062, 162149488, 253823178, 395698276

Offset: 0

Vaclav Kotesovec, Aug 24 2015

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..5000 (first 2001 terms from Vaclav Kotesovec)

Cf. A026007, A032302, A261561, A261563, A266857, A266891.

b:= proc(n, i) option remember;  `if`(n=0, 1, `if`(i<1, 0,
      add(2^j*binomial(i, j)*b(n-i*j, i-1), j=0..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..40);  # Alois P. Heinz, Sep 21 2018

nmax = 50; CoefficientList[Series[Product[(1 + 2*x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 50; CoefficientList[Series[Exp[Sum[(-1)^(k+1)*2^k/k*x^k/(1 - x^k)^2, {k, 1, nmax}]], {x, 0, nmax}], x]
nmax = 50; s = 1+2*x; Do[s*=Sum[Binomial[k, j]*2^j*x^(j*k), {j, 0, nmax/k}]; s = Take[Expand[s], Min[nmax + 1, Exponent[s, x] + 1]];, {k, 2, nmax}]; CoefficientList[s, x] (* Vaclav Kotesovec, Jan 08 2016 *)

{a(n) = polcoeff( exp( sum(m=1, n, x^m/m * sumdiv(m, d, -(-2)^d * m^2/d^2) ) +x*O(x^n)), n)}
for(n=0, 40, print1(a(n), ", ")) \\ Paul D. Hanna, Sep 30 2015

G.f.: exp( Sum_{n>=1} x^n/n * Sum_{d|n} -(-2)^d * n^2/d^2 ). - Paul D. Hanna, Sep 30 2015

a(n) ~ c^(1/6) * exp(3^(2/3)*c^(1/3)*n^(2/3)/2) / (3^(3/4)*sqrt(2*Pi)*n^(2/3)), where c = Pi^2*log(2) + log(2)^3 - 6*polylog(3, -1/2) = 10.00970018379942727227807189532511265744588249928680712064... . - Vaclav Kotesovec, Jan 04 2016

cp's OEIS Frontend

A299701 Number of distinct subset-sums of the integer partition with Heinz number n.

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A072706 Number of unimodal partitions/compositions of n into distinct terms.

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Maple

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A237258 Number of strict partitions of 2n that include a partition of n.

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A032308 Expansion of Product_{k>=1} (1 + 3*x^k).

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Maple

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A266891 Expansion of Product_{k>=1} (1 + k*x^k)^k.

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A316314 Number of distinct nonempty-subset-averages of the integer partition with Heinz number n.

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A108796 Number of unordered pairs of partitions of n (into distinct parts) with empty intersection.

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A261584 Expansion of Product_{k>=1} (1 + 2x^k)/(1 - 2x^k).