cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-8 of 8 results.

A027337 Number of partitions of n that do not contain 3 as a part.

Original entry on oeis.org

1, 1, 2, 2, 4, 5, 8, 10, 15, 19, 27, 34, 47, 59, 79, 99, 130, 162, 209, 259, 330, 407, 512, 628, 783, 956, 1181, 1435, 1760, 2129, 2594, 3124, 3784, 4539, 5468, 6534, 7834, 9327, 11132, 13208, 15701, 18568, 21989, 25923, 30592, 35960, 42297, 49579, 58139, 67967
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

a(n) is also the number of partitions of n with less than three 1's. - Geoffrey Critzer, Jun 20 2014

Crossrefs

Cf. A000041, A027336, A027338.
Column k=0 of A263232.
Column 3 of A175788.

Programs

  • Mathematica
    nn=49;CoefficientList[Series[(1-x^3)Product[1/(1-x^i),{i,1,nn}],{x,0,nn}],x] (* Geoffrey Critzer, Jun 20 2014 *)
  • PARI
    a(n)=if(n<0,0,polcoeff((1-x^3)/eta(x+x*O(x^n)),n))

Formula

G.f.: (1-x^3) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n) - A000041(n-3).
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 3*Pi/(2*sqrt(6)))/sqrt(n) + (37/8 + 9/(2*Pi^2) + 1801*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

A027338 Number of partitions of n that do not contain 4 as a part.

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 9, 12, 17, 23, 31, 41, 55, 71, 93, 120, 154, 196, 250, 314, 396, 495, 617, 765, 948, 1166, 1434, 1755, 2143, 2607, 3168, 3832, 4631, 5578, 6706, 8041, 9628, 11494, 13705, 16302, 19361, 22946, 27159, 32076, 37837, 44551, 52384, 61493
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Column 4 of A175788.

Programs

  • PARI
    a(n)=if(n<0,0,polcoeff((1-x^4)/eta(x+x*O(x^n)),n))

Formula

G.f.: (1-x^4) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-4).
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (3*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 4*Pi/(2*sqrt(6)))/sqrt(n) + (49/8 + 9/(2*Pi^2) + 3169*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

More terms from Benoit Cloitre, Dec 10 2002

A027340 Number of partitions of n that do not contain 6 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 10, 14, 20, 27, 37, 49, 66, 86, 113, 146, 189, 241, 308, 389, 492, 616, 771, 958, 1190, 1468, 1809, 2218, 2716, 3310, 4029, 4884, 5913, 7133, 8592, 10318, 12373, 14795, 17666, 21042, 25028, 29700, 35197, 41624, 49160, 57949, 68220
Offset: 0

Views

Author

Clark Kimberling

Keywords

Comments

Also number of partitions of n where no part appears more than five times.

Crossrefs

Column 6 of A175788.

Programs

  • PARI
    a(n)=if(n<0,0,polcoeff((1-x^6)/eta(x+x*O(x^n)),n))

Formula

G.f.: (1-x^6) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-6).
a(n) ~ Pi * exp(sqrt(2*n/3)*Pi) / (2*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 6*Pi/(2*sqrt(6)))/sqrt(n) + (73/8 + 9/(2*Pi^2) + 7057*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

More terms from Benoit Cloitre, Dec 10 2002

A027342 Number of partitions of n that do not contain 8 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 21, 29, 40, 53, 72, 94, 124, 161, 209, 267, 343, 434, 550, 691, 867, 1079, 1344, 1661, 2051, 2520, 3091, 3773, 4602, 5587, 6774, 8185, 9874, 11873, 14259, 17072, 20411, 24343, 28989, 34440, 40864, 48378, 57198, 67497, 79543
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Column 8 of A175788.

Programs

Formula

G.f.: (1-x^8) Product_{m>0} 1/(1-x^m).
a(n)=A000041(n)-A000041(n-8).
a(n) ~ 2*Pi * exp(sqrt(2*n/3)*Pi) / (3*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 8*Pi/(2*sqrt(6)))/sqrt(n) + (97/8 + 9/(2*Pi^2) + 12481*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

More terms from Benoit Cloitre, Dec 10 2002

A027343 Number of partitions of n that do not contain 9 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 29, 41, 54, 74, 96, 128, 165, 216, 275, 355, 448, 571, 715, 901, 1120, 1399, 1727, 2139, 2625, 3228, 3938, 4812, 5840, 7094, 8568, 10352, 12447, 14967, 17919, 21450, 25581, 30496, 36234, 43031, 50951, 60292
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

9th column of A175788. Cf. A000041, A027336, A027337-A027344.

Programs

Formula

G.f.: (1-x^9) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-9).
a(n) ~ 3*Pi * exp(sqrt(2*n/3)*Pi) / (4*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 9*Pi/(2*sqrt(6)))/sqrt(n) + (109/8 + 9/(2*Pi^2) + 15769*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

Edited by Alois P. Heinz, Dec 04 2010

A027344 Number of partitions of n that do not contain 10 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 41, 55, 75, 98, 130, 169, 220, 282, 363, 460, 585, 736, 925, 1154, 1440, 1782, 2205, 2713, 3333, 4075, 4977, 6050, 7347, 8888, 10735, 12925, 15541, 18627, 22297, 26620, 31734, 37741, 44825, 53118, 62865
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

10th column of A175788. Cf. A000041, A027336, A027337-A027343.

Programs

  • Maple
    A41:= n-> `if`(n<0, 0, combinat[numbpart](n)):
a:= n-> A41(n) -A41(n-10):
seq(a(n), n=0..50);

Formula

G.f.: (1-x^10) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-10).
a(n) ~ 5*Pi * exp(sqrt(2*n/3)*Pi) / (6*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 10*Pi/(2*sqrt(6)))/sqrt(n) + (121/8 + 9/(2*Pi^2) + 19441*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

More terms from Benoit Cloitre, Dec 10 2002
Edited by Alois P. Heinz, Dec 04 2010

A027339 Number of partitions of n that do not contain 5 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 10, 13, 19, 25, 35, 45, 62, 79, 105, 134, 175, 220, 284, 355, 451, 561, 705, 870, 1085, 1331, 1644, 2008, 2463, 2990, 3646, 4406, 5339, 6425, 7745, 9279, 11135, 13288, 15872, 18875, 22455, 26606, 31537, 37246, 43990, 51796, 60975
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Column 5 of A175788.

Programs

Formula

G.f.: (1-x^5) Product_{m>0} 1/(1-x^m).
a(n) = A000041(n)-A000041(n-5).
a(n) ~ 5*Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 5*Pi/(2*sqrt(6)))/sqrt(n) + (61/8 + 9/(2*Pi^2) + 4921*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

More terms from Benoit Cloitre, Dec 10 2002

A027341 Number of partitions of n that do not contain 7 as a part.

Original entry on oeis.org

1, 1, 2, 3, 5, 7, 11, 14, 21, 28, 39, 51, 70, 90, 120, 154, 201, 255, 329, 413, 526, 657, 826, 1024, 1278, 1573, 1946, 2383, 2926, 3563, 4349, 5267, 6391, 7707, 9300, 11165, 13412, 16033, 19173, 22836, 27195, 32273, 38291, 45284, 53538, 63119, 74373
Offset: 0

Views

Author

Clark Kimberling

Keywords

Crossrefs

Column 7 of A175788.

Programs

Formula

G.f.: (1-x^7) Product_{m>0} 1/(1-x^m).
a(n)=A000041(n)-A000041(n-7).
a(n) ~ 7*Pi * exp(sqrt(2*n/3)*Pi) / (12*sqrt(2)*n^(3/2)) * (1 - (3*sqrt(3/2)/Pi + Pi/(24*sqrt(6)) + 7*Pi/(2*sqrt(6)))/sqrt(n) + (85/8 + 9/(2*Pi^2) + 9577*Pi^2/6912)/n). - Vaclav Kotesovec, Nov 04 2016

Extensions

More terms from Benoit Cloitre, Dec 10 2002
Showing 1-8 of 8 results.

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