A306099 -id:A306099 - cp's OEIS frontend

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.

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

M. F. Hasler, Sep 22 2018

			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 ...]

Alois P. Heinz, Antidiagonals n = 0..50, flattened
OEIS wiki, Plane partitions.
Wikipedia, Plane partition.

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

A306100(n,k)=sum(j=1,n,A091298(n,j)*k^j)

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

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

M. F. Hasler, Sep 22 2018

nonn

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.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A091298, A208447.

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

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

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

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

A306101 Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n, k >= 1; read by antidiagonals.

Original entry on oeis.org

1, 2, 3, 3, 10, 6, 4, 21, 34, 13, 5, 36, 102, 122, 24, 6, 55, 228, 525, 378, 48, 7, 78, 430, 1540, 2334, 1242, 86, 8, 105, 726, 3605, 8964, 11100, 3690, 160, 9, 136, 1134, 7278, 25980, 56292, 47496, 11266, 282, 10, 171, 1672, 13237, 62574, 203280, 316388, 210756, 32666, 500, 11, 210, 2358, 22280, 132258, 586878, 1417530

Offset: 1

M. F. Hasler, Sep 22 2018

One could have included a row 0 with all 1's, since there is exactly one partition of n = 0, the empty sum, for which all terms (since there are none) are colored in one among k colors.

			The array starts:
  [      1       2       3       4       5 ...] = A000027
  [      3      10      21      36      55 ...] = A014105
  [      6      34     102     228     430 ...] = A067389
  [     13     122     525    1540    3605 ...]
  [     24     378    2334    8964   25980 ...]
  [     48    1242   11100   56292  203280 ...]
   A000219 A306099 A306093 A306094 A306094
For concrete examples, see A306099 and A306093.

Alois P. Heinz, Antidiagonals n = 1..50, flattened

Cf. A091298, A208447, A000219, A000027, A014105, A067389.
See A306100 for a variant.
Cf. A000219, A306099, A306093, A306094, A306095 for columns 1..5.

A306101(n,k)=sum(j=1,n,A091298(n,j)*k^j)

T(n,k) = Sum_{j=1..n} A091298(n,j)*k^j.

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

M. F. Hasler, Sep 22 2018

nonn

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.

			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.

Alois P. Heinz, Table of n, a(n) for n = 0..50

Cf. A091298, A208447.

Column 4 of A306100 and A306101. See A306099 and A306093 for columns 2 and 3.

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

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

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

M. F. Hasler, Sep 22 2018

nonn

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.

			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.

M. F. Hasler, Table of n, a(n) for n = 0..50

Cf. A091298, A208447.

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

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

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

M. F. Hasler, Oct 16 2018

nonn

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.

			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.

M. F. Hasler, Table of n, a(n) for n = 0..50

Cf. A091298, A208447.

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

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

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

cp's OEIS Frontend

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.

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

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A306101 Square array T(n,k) = number of plane partitions of n with parts colored in (at most) k colors; n, k >= 1; read by antidiagonals.

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

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

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

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PARI

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