A000219 -id:A000219 - cp's OEIS frontend

A161870 Convolution square of A000219.

1, 2, 7, 18, 47, 110, 258, 568, 1237, 2600, 5380, 10870, 21652, 42350, 81778, 155676, 292964, 544846, 1003078, 1828128, 3301952, 5911740, 10499385, 18502582, 32371011, 56240816, 97073055, 166497412, 283870383, 481212656, 811287037, 1360575284, 2270274785, 3769835178, 6230705170, 10251665550, 16794445441

Offset: 0

Gary W. Adamson, Jun 20 2009

nonn

Equals [1,2,3,...] * [1,0,4,0,10,0,20,...] * [1,0,0,6,0,0,21,...] * [1,0,0,0,8,0,0,0,36,...] * ... - Gary W. Adamson, Jul 06 2009

Number of pairs of planar partitions of u and v where u + v = n. - Joerg Arndt, Apr 22 2014

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from Vaclav Kotesovec)
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015, p. 19.
Paul Martin, Eric C. Rowell, and Fiona Torzewska, Classification of charge-conserving loop braid representations, arXiv:2301.13831 [math.QA], 2023.

Cf. A000219.

Column k=2 of A255961.

a:= proc(n) option remember; `if`(n=0, 1, 2*add(
      a(n-j)*numtheory[sigma][2](j), j=1..n)/n)
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 12 2015

nn = 36; CoefficientList[Series[Product[1/(1 - x^i)^(2 i), {i, 1, nn}] , {x, 0, nn}], x] (* Geoffrey Critzer, Nov 29 2014 *)

N=66;x='x+O('x^N); Vec(1/prod(k=1,N,(1-x^k)^k)^2) \\ Joerg Arndt, Apr 22 2014

G.f.: 1 / prod(k>=1, (1-x^k)^k )^2. - Joerg Arndt, Apr 22 2014
a(n) ~ Zeta(3)^(2/9) * exp(1/6 + 3*n^(2/3)*(Zeta(3)/2)^(1/3)) / (A^2 * 2^(1/18) * sqrt(3*Pi) * n^(13/18)), where A = A074962 = 1.2824271291... is the Glaisher-Kinkelin constant and Zeta(3) = A002117 = 1.202056903... . - Vaclav Kotesovec, Feb 27 2015
G.f.: exp(2*Sum_{k>=1} x^k/(k*(1 - x^k)^2)). - Ilya Gutkovskiy, May 29 2018

Added more terms, Joerg Arndt, Apr 22 2014

A091360 Partial sums of A000219.

Original entry on oeis.org

1, 2, 5, 11, 24, 48, 96, 182, 342, 624, 1124, 1983, 3462, 5947, 10114, 16993, 28290, 46624, 76225, 123555, 198833, 317627, 504102, 794885, 1246079, 1942112, 3010857, 4643515, 7126749, 10886361, 16555324, 25067633, 37801062, 56776035, 84951990, 126643036, 188127997, 278507781, 410949776, 604437277, 886284200, 1295668181

Offset: 0

Christian G. Bower, Jan 02 2004

nonn

Convergent of columns of A091355.

Joerg Arndt and Vaclav Kotesovec, Table of n, a(n) for n = 0..5000 (first 500 terms from Joerg Arndt)
N. J. A. Sloane, Transforms

Cf. A000219, A002117, A074962.

CoefficientList[Series[1/(1-x)*Product[1/(1-x^k)^k,{k,1,50}],{x,0,50}],x] (* Vaclav Kotesovec, Aug 16 2015 *)

N=66; x='x+O('x^N); Vec( 1/((1-x)*prod(n=1,N, (1-x^n)^n )) ) \\ Joerg Arndt, Mar 15 2014

Euler transform of 2, 2, 3, 4, 5, 6, 7, 8, 9, ...
G.f.: 1/( (1-x) * prod(n>=1, (1-x^n)^n ) ). [Joerg Arndt, Mar 15 2014]
From Vaclav Kotesovec, Aug 16 2015: (Start)
a(n) = Sum_{k=0..n} A000219(k).
a(n) ~ (n/(2*Zeta(3)))^(1/3) * A000219(n).
a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(2/3)) / (A * 2^(23/36) * sqrt(3*Pi) * Zeta(3)^(5/36) * n^(13/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.
(End)
G.f.: exp(Sum_{k>=1} (sigma_2(k) + 1)*x^k/k). - Ilya Gutkovskiy, Aug 21 2018

A048141 Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have a threefold axis of symmetry that is the intersection of 3 mirror planes, i.e., C3v symmetry.

Original entry on oeis.org

1, 0, 0, 1, 0, 0, 2, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 4, 4, 0, 4, 5, 0, 5, 7, 1, 6, 9, 1, 6, 11, 1, 8, 15, 2, 10, 20, 3, 10, 25, 4, 12, 33, 7, 14, 40, 9, 15, 48, 12, 18, 60, 17, 20, 74, 23, 22, 89, 30, 26, 108, 40, 30, 130, 51, 33, 157, 66, 37, 187, 85, 42, 222, 108, 47, 262, 136, 54

Offset: 1

Wouter Meeussen

			The plane partition {{2,1},{1}} has C3v symmetry.

Wouter Meeussen, Table of n, a(n) for n = 1..202

Cf. A000219, A000784, A000785, A000786, A005987, A048142, A051056-A051061, A096419.

A048142 Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have only a threefold axis of symmetry, i.e., C3 symmetry.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 1, 1, 0, 2, 2, 0, 3, 3, 0, 5, 6, 0, 7, 9, 0, 11, 16, 1, 14, 23, 2, 20, 36, 4, 27, 52, 7, 37, 78, 13, 48, 111, 21, 65, 163, 36, 83, 227, 56, 109, 322, 89, 139, 444, 135, 179, 618, 207, 226, 841, 305, 288, 1151, 453, 361

Offset: 1

Wouter Meeussen

Only one of the pair left-handed, right-handed, is enumerated.

			The plane partitions {{3, 2, 2}, {3, 1}, {1, 1}} and {{3, 2, 2}, {3, 2}, {1, 1}} have C3 symmetry.

Wouter Meeussen, Table of n, a(n) for n = 1..202

Cf. A000219, A000784, A000785, A000786, A005987, A048141, A051056-A051061, A096419.

a(n) = (A096419(n) - A048141(n))/2.

A191659 First differences of A000219.

Original entry on oeis.org

0, 2, 3, 7, 11, 24, 38, 74, 122, 218, 359, 620, 1006, 1682, 2712, 4418, 7037, 11267, 17729, 27948, 43516, 67681, 104308, 160411, 244839, 372712, 563913, 850576, 1276378, 1909351, 2843346, 4221120, 6241544, 9200982, 13515091, 19793915, 28894823, 42062211, 61045506, 88359422, 127537058, 183617286, 263666228, 377696338, 539715276, 769456793

Offset: 0

N. J. A. Sloane, Jun 10 2011

nonn

G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.

Cf. A000219, A191660, A191661.

nmax = 60; Rest[CoefficientList[Series[Product[1/(1 - x^k)^k, {k, 2, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Oct 05 2015 *)

a(n) ~ 2^(1/36) * Zeta(3)^(19/36) * exp(1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2^(2/3)) / (A * sqrt(3*Pi) * n^(37/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 05 2015

A000785 Number of asymmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have no symmetry.

Original entry on oeis.org

0, 0, 0, 1, 2, 5, 11, 21, 39, 73, 129, 226, 388, 659, 1100, 1821, 2976, 4828, 7754, 12370, 19574, 30789, 48097, 74725, 115410, 177366, 271159, 412665, 625098, 942932, 1416362, 2119282, 3158840, 4691431, 6942882, 10240503, 15054705

Offset: 1

N. J. A. Sloane

P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-François Alcover, Table of n, a(n) for n = 1..150
P. A. MacMahon, Combinatory analysis.

Equals (A048141 - 3*A048140 + 2*A000219 - A048142)/3.

Cf. A000784, A000786, A005987.

nmax = 150;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2 i - 1))/(1 - x^(2 i))^Floor[i/2], {i, 1, Ceiling[( nmax + 1)/2]}] + O[x]^( nmax + 1);
A005987 = CoefficientList[s, x];
a048140[n_] := (a219[n] + A005987[[n + 1]])/2;
A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {, }][[All, 2]];
A048142 = Cases[Import["https://oeis.org/A048142/b048142.txt", "Table"], {, }][[All, 2]];
a[1] = 0;
a[n_] := (A048141[[n]] - 3 a048140[n] + 2 a219[n] - A048142[[n]])/3;
a /@ Range[1, nmax] (* Jean-François Alcover, Dec 28 2019 *)

More terms from Wouter Meeussen

A294500 Binomial transform of the number of planar partitions (A000219).

Original entry on oeis.org

1, 2, 6, 19, 60, 185, 559, 1662, 4875, 14134, 40564, 115370, 325465, 911355, 2534595, 7004827, 19246626, 52596377, 143006632, 386984573, 1042537831, 2796803110, 7473161196, 19893461042, 52767059608, 139488323734, 367540167625, 965445514862, 2528516552660

Offset: 0

Vaclav Kotesovec, Nov 01 2017

nonn

Let 0 < p < 1, r > 0, v > 0, f(n) = v*exp(r*n^p)/n^b, then
Sum_{k=0..n} binomial(n,k) * f(k) ~ f(n/2) * 2^n * exp(g(n)), where
g(n) = p^2 * r^2 * n^p / (2^(1+2*p)*n^(1-p) + p*r*(1-p)*2^(1+p)).
Special cases:
p < 1/2, g(n) = 0
p = 1/2, g(n) = r^2/16
p = 2/3, g(n) = r^2 * n^(1/3) / (9 * 2^(1/3)) - r^3/81
p = 3/4, g(n) = 9*r^2*sqrt(n)/(64*sqrt(2)) - 27*r^3*n^(1/4)/(2048*2^(1/4)) + 81*r^4/65536
p = 3/5, g(n) = 9*r^2*n^(1/5)/(100*2^(1/5))
p = 4/5, g(n) = 2^(7/5)*r^2*n^(3/5)/25 - 4*2^(3/5)*r^3*n^(2/5)/625 + 8*2^(4/5)*r^4*n^(1/5)/15625 - 32*r^5/390625

Vaclav Kotesovec, Table of n, a(n) for n = 0..2930

Cf. A218481, A266232, A294501, A294502, A294504.

nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} binomial(n,k) * A000219(k).
a(n) ~ exp(1/12 + 3 * Zeta(3)^(1/3) * n^(2/3) / 2^(4/3) + Zeta(3)^(2/3) * n^(1/3) / 2^(5/3) - Zeta(3)/12) * 2^(n + 7/18) * Zeta(3)^(7/36) / (A * sqrt(3*Pi) * n^(25/36)), where A is the Glaisher-Kinkelin constant A074962.
G.f.: (1/(1 - x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 - x)^k)). - Ilya Gutkovskiy, Aug 20 2018

A000784 Number of symmetrical planar partitions of n (planar partitions (A000219) that when regarded as 3-D objects have just one symmetry plane).

Original entry on oeis.org

0, 1, 2, 2, 4, 6, 6, 11, 16, 20, 28, 41, 51, 70, 93, 122, 158, 211, 266, 350, 450, 577, 730, 948, 1186, 1510, 1901, 2408, 2999, 3790, 4703, 5898, 7310, 9111, 11231, 13979, 17168, 21229, 26036, 32095, 39188, 48155, 58657, 71798, 87262, 106472, 129014

Offset: 1

N. J. A. Sloane

P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Jean-François Alcover, Table of n, a(n) for n = 1..150
P. A. MacMahon, Combinatory analysis.

Equals -A048141 + 2*A048140 - A000219.

Cf. A000785, A000786, A005987, A048142.

nmax = 150;
a219[0] = 1;
a219[n_] := a219[n] = Sum[a219[n - j] DivisorSigma[2, j], {j, n}]/n;
s = Product[1/(1 - x^(2 i - 1))/(1 - x^(2 i))^Floor[i/2], {i, 1, Ceiling[( nmax + 1)/2]}] + O[x]^( nmax + 1);
A005987 = CoefficientList[s, x];
a048140[n_] := (a219[n] + A005987[[n + 1]])/2;
A048141 = Cases[Import["https://oeis.org/A048141/b048141.txt", "Table"], {, }][[All, 2]];
a[1] = 0;
a[n_] := -A048141[[n]] + 2 a048140[n] - a219[n];
a /@ Range[1, nmax] (* Jean-François Alcover, Dec 28 2019 *)

More terms from Wouter Meeussen

A294501 Inverse binomial transform of the number of planar partitions (A000219).

Original entry on oeis.org

1, 0, 2, -1, 4, -7, 19, -48, 123, -304, 728, -1694, 3865, -8735, 19739, -44875, 102818, -236939, 546988, -1260023, 2888607, -6584008, 14927816, -33714166, 75976024, -171095098, 385405617, -868708176, 1959010348, -4417777937, 9957188242, -22420045445

Offset: 0

Vaclav Kotesovec, Nov 01 2017

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..3000

Cf. A281425, A293467, A294500, A294503, A294505.

nmax = 40; s = CoefficientList[Series[Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]; Table[Sum[(-1)^(n-k) * Binomial[n, k] * s[[k+1]], {k, 0, n}], {n, 0, nmax}]

a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(n,k) * A000219(k).

G.f.: (1/(1 + x))*exp(Sum_{k>=1} sigma_2(k)*x^k/(k*(1 + x)^k)). - Ilya Gutkovskiy, Aug 20 2018

A323584 Second Moebius transform of A000219. Number of plane partitions of n whose multiset of rows is aperiodic and whose multiset of columns is also aperiodic.

Original entry on oeis.org

1, 1, 1, 4, 8, 22, 34, 84, 137, 271, 450, 857, 1373, 2483, 3993, 6823, 10990, 18332, 28966, 47328, 74286, 118614, 184755, 290781, 448010, 695986, 1063773, 1632100, 2474970, 3759610, 5654233, 8512307, 12710995, 18973247, 28139285, 41690830, 61423271, 90379782

Offset: 0

Gus Wiseman, Jan 19 2019

nonn

A multiset is aperiodic if its multiplicities are relatively prime.

Also the number of plane partitions of n whose multiset of rows is aperiodic and whose parts are relatively prime.

			The a(4) = 8 plane partitions with aperiodic multisets of rows and columns:
  4   31   211
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1
The a(4) = 8 plane partitions with aperiodic multiset of rows and relatively prime parts:
  31   211   1111
.
  3   21   111
  1   1    1
.
  2   11
  1   1
  1   1

Alois P. Heinz, Table of n, a(n) for n = 0..10000

Cf. A000219, A000837, A003293, A100953, A300275, A303546, A320802, A321390, A323585, A323587.

primeMS[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
ptnplane[n_]:=Union[Map[Reverse@*primeMS,Join@@Permutations/@facs[n],{2}]];
Table[Sum[Length[Select[ptnplane[Times@@Prime/@y],And[GCD@@Length/@Split[#]==1,And@@GreaterEqual@@@#,And@@(GreaterEqual@@@Transpose[PadRight[#]])]&]],{y,Select[IntegerPartitions[n],GCD@@#==1&]}],{n,10}]

The Moebius transform T of a sequence q is T(q)(n) = Sum_{d|n} mu(n/d) * q(d) where mu = A008683. The first Moebius transform of A000219 is A300275 and the third is A323585.

cp's OEIS Frontend

A161870 Convolution square of A000219.

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A091360 Partial sums of A000219.

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A048141 Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have a threefold axis of symmetry that is the intersection of 3 mirror planes, i.e., C3v symmetry.

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A048142 Number of symmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have only a threefold axis of symmetry, i.e., C3 symmetry.

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A191659 First differences of A000219.

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A000785 Number of asymmetrical planar partitions of n: planar partitions (A000219) that when regarded as 3-D objects have no symmetry.

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A294500 Binomial transform of the number of planar partitions (A000219).

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A000784 Number of symmetrical planar partitions of n (planar partitions (A000219) that when regarded as 3-D objects have just one symmetry plane).

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