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 1-3 of 3 results.

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.
		

Crossrefs

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.
		

Crossrefs

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

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.
		

Crossrefs

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.
Showing 1-3 of 3 results.