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 1-6 of 6 results.

A306100 Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n >= 0, k >= 0; read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, 3, 0, 1, 3, 10, 6, 0, 1, 4, 21, 34, 13, 0, 1, 5, 36, 102, 122, 24, 0, 1, 6, 55, 228, 525, 378, 48, 0, 1, 7, 78, 430, 1540, 2334, 1242, 86, 0, 1, 8, 105, 726, 3605, 8964, 11100, 3690, 160, 0, 1, 9, 136, 1134, 7278, 25980, 56292, 47496, 11266, 282, 0
Offset: 0

Views

Author

M. F. Hasler, Sep 22 2018

Keywords

Examples

			The array starts:
  [1  1    1     1     1      1 ...] = A000012
  [0  1    2     3     4      5 ...] = A001477
  [0  3   10    21    36     55 ...] = A014105
  [0  6   34   102   228    430 ...] = A067389
  [0 13  122   525  1540   3605 ...]
  [0 24  378  2334  8964  25980 ...]
  [0 48 1242 11100 56292 203280 ...]

Links

Crossrefs

Columns k=0-5 give: A000007, A000219, A306099, A306093, A306094, A306095.
See A306101 for a variant.
Cf. A091298, A208447, A001477, A014105, A067389.

Programs

Formula

T(n,k) = Sum_{j=0..n} A091298(n,j)*k^j, assuming A091298(n,0) = A000007(n).
T(n,k) = Sum_{i=0..k} C(k,i) * A319600(n,i). - Alois P. Heinz, Sep 28 2018

Extensions

Edited by Alois P. Heinz, Sep 26 2018

A306093 Number of plane partitions of n where parts are colored in 3 colors.

Original entry on oeis.org

1, 3, 21, 102, 525, 2334, 11100, 47496, 210756, 886080, 3759114, 15378051, 63685767, 255417357, 1030081827, 4078689249, 16150234665, 62991117084, 245948154087, 947944122906, 3653360869998, 13946363438502, 53149517598207, 200994216333375, 759191650345380
Offset: 0

Views

Author

M. F. Hasler, Sep 22 2018

Keywords

Comments

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among three given colors, since there is no term at all.

Examples

			For n = 1, there is only the partition [1], which can be colored in any of the three colors, whence a(1) = 3.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 3 + 9 + 9 = 21 distinct possibilities.

Links

Crossrefs

Cf. A091298, A208447.
Column 3 of A306100 and A306101. See A306099 for column 2, A306094 .. A306096 for columns 4 .. 6.

Programs

  • PARI
    a(n)=sum(k=1,n,A091298(n,k)*3^k,!n)

Formula

a(n) = Sum_{k=1..n} A091298(n,k)*3^k.

Extensions

a(12) corrected and a(13)-a(24) added by Alois P. Heinz, Sep 24 2018

A306099 Number of plane partitions of n where parts are colored in 2 colors.

Original entry on oeis.org

1, 2, 10, 34, 122, 378, 1242, 3690, 11266, 32666, 94994, 267202, 754546, 2072578, 5691514, 15364290, 41321962, 109634586, 290048746, 758630698, 1977954706, 5111900410, 13161995010, 33645284962, 85727394018, 217042978882, 547750831210, 1375147078146, 3441516792442
Offset: 0

Views

Author

M. F. Hasler and Rick L. Shepherd, following an idea from David S. Newman, Sep 22 2018

Keywords

Comments

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among two given colors, since there is no term at all.

Examples

			For n = 1, there is only the partition [1], which can be colored in any of the two colors, whence a(1) = 2.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 2 + 4 + 4 = 10 distinct possibilities.

Links

Crossrefs

Cf. A091298, A208447.
Column 2 of A306100 and A306101. See A306093 .. A306096 for columns 3 .. 6.

Programs

  • PARI
    a(n)=!n+sum(k=1,n,A091298(n,k)<

Formula

a(n) = Sum_{k=1..n} A091298(n,k)*2^k.

Extensions

a(12) corrected and a(13)-a(28) added by Alois P. Heinz, Sep 24 2018

A306094 Number of plane partitions of n where parts are colored in (at most) 4 colors.

Original entry on oeis.org

1, 4, 36, 228, 1540, 8964, 56292, 316388, 1857028, 10301892, 57884132, 312915172, 1720407492, 9132560068, 48898964964, 256790538660, 1350883911620, 6992031608260, 36296271612324, 185785685287076, 952221494828996, 4831039856692356, 24489621255994276
Offset: 0

Views

Author

M. F. Hasler, Sep 22 2018

Keywords

Comments

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among four given colors, since there is no term at all.

Examples

			For n = 1, there is only the partition [1], which can be colored in any of the four colors, whence a(1) = 4.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 4 + 16 + 16 = 36 distinct possibilities.

Links

Crossrefs

Cf. A091298, A208447.
Column 4 of A306100 and A306101. See A306099 and A306093 for columns 2 and 3.

Programs

  • PARI
    a(n)=!n+sum(k=1,n,A091298(n,k)*4^k)

Formula

a(n) = Sum_{k=1..n} A091298(n,k)*4^k.

Extensions

a(12) corrected and a(13)-a(22) added by Alois P. Heinz, Sep 24 2018

A306095 Number of plane partitions of n where parts are colored in (at most) 5 colors.

Original entry on oeis.org

1, 5, 55, 430, 3605, 25980, 203280, 1417530, 10373080, 71595830, 501688880, 3376856755, 23181027055, 153326091805, 1024829902855, 6713038952355, 44092634675905, 284723995000530, 1845944380173205, 11791816763005330, 75485171060740630, 478105767714603130
Offset: 0

Views

Author

M. F. Hasler, Sep 22 2018

Keywords

Comments

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among five given colors, since there is no term at all.

Examples

			For n = 1, there is only the partition [1], which can be colored in any of the five colors, whence a(1) = 5.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 5 + 25 + 25 = 55 distinct possibilities.

Links

Crossrefs

Cf. A091298, A208447.
Column 5 of A306100 and A306101. See A306099, A306093, A306094, A306096 for columns 2, 3, 4 and 6.

Programs

  • PARI
    a(n)=!n+sum(k=1,n,A091298(n,k)*5^k)

Formula

a(n) = Sum_{k=1..n} A091298(n,k)*5^k.

A306096 Number of plane partitions of n where parts are colored in (at most) 6 colors.

Original entry on oeis.org

1, 6, 78, 726, 7278, 62574, 586878, 4889166, 42892710, 354335982, 2976581670, 23990771094, 197564663094, 1565310230790, 12548473437822, 98526949264374, 776195574339102, 6008457242324814, 46729763436714126, 357901583160822990, 2748384845416097718
Offset: 0

Views

Author

M. F. Hasler, Oct 16 2018

Keywords

Comments

a(0) = 1 corresponds to the empty sum, in which all terms are colored in one among six given colors, since there is no term at all.

Examples

			For n = 1, there is only the partition [1], which can be colored in any of the six colors, whence a(1) = 6.
For n = 2, there are the partitions [2], [1,1] and [1;1]. Adding colors, this yields a(2) = 6 + 36 + 36 = 78 distinct possibilities.

Links

Crossrefs

Cf. A091298, A208447.
Column 6 of A306100 and A306101. See A306099, A306093, A306094, A306095 for columns 2, 3, 4 and 5.

Programs

  • PARI
    a(n)=sum(k=1,n,A091298(n,k)*6^k,!n)

Formula

a(n) = Sum_{k=1..n} A091298(n,k)*6^k, for n > 0.
Showing 1-6 of 6 results.

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