A266647 -id:A266647 - cp's OEIS frontend

A100405 Number of partitions of n where every part appears more than two times.

1, 0, 0, 1, 1, 1, 2, 1, 2, 3, 3, 3, 7, 5, 6, 11, 10, 10, 17, 15, 20, 26, 25, 29, 44, 41, 47, 63, 67, 72, 99, 97, 114, 143, 148, 168, 216, 216, 248, 306, 328, 358, 443, 462, 527, 629, 665, 739, 898, 936, 1055, 1238, 1330, 1465, 1727, 1837, 2055, 2366, 2543, 2808, 3274

Offset: 0

Vladeta Jovovic, Jan 11 2005

			a(6)=2 because we have [2,2,2] and [1,1,1,1,1,1].

Alois P. Heinz, Table of n, a(n) for n = 0..10000 (terms n = 1..5000 from Vaclav Kotesovec)

Cf. A007690, A160974-A160990.
Cf. A266647, A266648, A266649, A266650.
Cf. A161294.

G:=product((1+x^(3*k)/(1-x^k)),k=1..30): Gser:=series(G,x=0,80): seq(coeff(Gser,x,n),n=0..70); # Emeric Deutsch, Aug 06 2005
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      add(b(n-i*j, i-1), j=[0, $3..iquo(n, i)])))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..70);  # Alois P. Heinz, Aug 20 2019

nmax = 100; Rest[CoefficientList[Series[Product[1 + x^(3*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Nov 28 2015 *)

G.f.: Product_{k>0} (1+x^(3*k)/(1-x^k)). More generally, g.f. for number of partitions of n where every part appears more than m times is Product_{k>0} (1+x^((m+1)*k)/(1-x^k)).

a(n) ~ sqrt(Pi^2 + 6*c) * exp(sqrt((2*Pi^2/3 + 4*c)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 - exp(-x) + exp(-3*x)) dx = -0.77271248407593487127235205445116662610863126869049971822566... . - Vaclav Kotesovec, Jan 05 2016

More terms from Emeric Deutsch, Aug 06 2005

a(0)=1 prepended by Alois P. Heinz, Aug 20 2019

A266648 Expansion of Product_{k>=1} (1 + x^(3*k)) / (1 - x^k).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 15, 21, 31, 46, 64, 89, 126, 170, 231, 314, 417, 552, 733, 955, 1244, 1617, 2079, 2665, 3413, 4331, 5485, 6931, 8704, 10901, 13629, 16949, 21033, 26045, 32123, 39529, 48553, 59429, 72599, 88518, 107624, 130599, 158209, 191175, 230611, 277717, 333730, 400375, 479598, 573386, 684481

Offset: 0

Vaclav Kotesovec, Jan 02 2016

nonn

a(n) is the number of overpartitions wherein only parts that are a multiple of three may be overlined. - Alois P. Heinz, Feb 03 2025

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Wenjie Fang, Hsien-Kuei Hwang, and Mihyun Kang, Phase transitions from exp(n^(1/2)) to exp(n^(2/3)) in the asymptotics of banded plane partitions, arXiv:2004.08901 [math.CO], 2020, p. 6.
Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.

Cf. A000726, A015128, A100405, A266647, A266649, A266650, A285445, A285447.

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(irem(i, 3)=0, 2, 1)*add(b(n-i*j, i-1), j=1..n/i)))
    end:
a:= n-> b(n$2):
seq(a(n), n=0..50);  # Alois P. Heinz, Feb 03 2025

nmax = 50; CoefficientList[Series[Product[(1+x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(7) * exp(sqrt(7*n)*Pi/3) / (24*n).

A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

Original entry on oeis.org

1, 1, 1, 3, 2, 4, 6, 7, 8, 13, 16, 18, 26, 29, 38, 49, 58, 68, 90, 101, 125, 156, 181, 214, 263, 304, 358, 435, 505, 589, 701, 812, 939, 1115, 1275, 1485, 1736, 1991, 2286, 2667, 3038, 3489, 4028, 4588, 5240, 6036, 6833, 7787, 8904, 10078, 11429, 13020, 14698

Offset: 0

Vaclav Kotesovec, Nov 28 2015

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000009, A000726, A001935, A100405, A266647, A266648, A266649, A266650, A266686, A275820.

nmax = 100; CoefficientList[Series[Product[1+x^k+x^(3*k), {k,1,nmax}], {x,0,nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = 1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] + If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(3*Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) + exp(-3*x)) dx = 0.9953865985263189816963357718655148864441174218433250148867... . - Vaclav Kotesovec, Jan 05 2016

A266686 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)).

Original entry on oeis.org

1, 1, 1, 1, 2, 2, 2, 3, 4, 5, 4, 6, 8, 9, 10, 11, 14, 16, 18, 21, 25, 28, 31, 36, 41, 48, 52, 59, 69, 77, 85, 96, 109, 121, 133, 151, 172, 189, 208, 231, 260, 287, 316, 350, 390, 432, 471, 521, 578, 636, 695, 764, 842, 924, 1009, 1107, 1218, 1330, 1449, 1584

Offset: 0

Vaclav Kotesovec, Jan 02 2016

nonn

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A264905, A266647, A266650, A275820.

nmax=60; CoefficientList[Series[Product[1+x^k-x^(3*k), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 100; p = ConstantArray[0, nmax + 1]; p[[1]] = 1; p[[2]] = 1; p[[4]] = -1; Do[Do[p[[j+1]] = p[[j+1]] + p[[j - k + 1]] - If[j < 3*k, 0, p[[j - 3*k + 1]]], {j, nmax, k, -1}];, {k, 2, nmax}]; p (* Vaclav Kotesovec, May 10 2018 *)

a(n) ~ c^(1/4) * exp(2*sqrt(c*n)) / (2*sqrt(Pi)*n^(3/4)), where c = Integral_{0..infinity} log(1 + exp(-x) - exp(-3*x)) dx = 0.59698046904738615106237970379036510874974380079287087827737... . - Vaclav Kotesovec, Jan 05 2016

A266650 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)) / (1 - x^k).

Original entry on oeis.org

1, 2, 4, 7, 13, 21, 34, 53, 82, 123, 181, 263, 379, 537, 754, 1047, 1444, 1972, 2675, 3601, 4820, 6408, 8473, 11141, 14580, 18985, 24611, 31765, 40839, 52294, 66719, 84819, 107474, 135731, 170892, 214518, 268524, 335190, 417308, 518212, 641948, 793324, 978157

Offset: 0

Vaclav Kotesovec, Jan 02 2016

nonn

Convolution of A266686 and A000041.

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A100405, A266647, A266648, A266649, A266686.

nmax = 50; CoefficientList[Series[Product[(1+x^k-x^(3*k))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

a(n) ~ sqrt(6*c + Pi^2) * exp(sqrt((4*c + 2*Pi^2/3)*n)) / (4*sqrt(3)*Pi*n), where c = Integral_{0..infinity} log(1 + exp(-x) - exp(-3*x)) dx = 0.59698046904738615106237970379036510874974380079287087827737... . - Vaclav Kotesovec, Jan 05 2016

A266649 Expansion of Product_{k>=1} 1 - x^(3*k)/(1-x^k).

Original entry on oeis.org

1, 0, 0, -1, -1, -1, -2, -1, -2, -1, -1, 1, -1, 3, 2, 5, 4, 8, 7, 11, 8, 12, 11, 13, 10, 9, 9, 7, 3, -2, -5, -13, -16, -25, -28, -48, -44, -66, -60, -82, -82, -104, -95, -120, -103, -131, -107, -133, -98, -124, -85, -94, -42, -51, 9, 7, 83, 100, 181, 208, 298

Offset: 0

Vaclav Kotesovec, Jan 02 2016

sign

Vaclav Kotesovec, Table of n, a(n) for n = 0..5000

Cf. A000726, A100405, A266647, A266648, A266650.

nmax = 100; CoefficientList[Series[Product[1-x^(3*k)/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} (1 - x^k - x^(3*k)) / (1 - x^k).

cp's OEIS Frontend

A100405 Number of partitions of n where every part appears more than two times.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A266648 Expansion of Product_{k>=1} (1 + x^(3*k)) / (1 - x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A264905 Expansion of Product_{k>=1} (1 + x^k + x^(3*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A266686 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula

A266650 Expansion of Product_{k>=1} (1 + x^k - x^(3*k)) / (1 - x^k).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A266649 Expansion of Product_{k>=1} 1 - x^(3*k)/(1-x^k).

Views

Author

Keywords

Links

Crossrefs

Programs

Mathematica

Formula