|
| |
|
|
A000293
|
|
a(n) = number of solid (i.e. three-dimensional) partitions of n.
(Formerly M3392 N1371)
|
|
30
| |
|
|
1, 1, 4, 10, 26, 59, 140, 307, 684, 1464, 3122, 6500, 13426, 27248, 54804, 108802, 214071, 416849, 805124, 1541637, 2930329, 5528733, 10362312, 19295226, 35713454, 65715094, 120256653, 218893580, 396418699, 714399381, 1281403841, 2287986987, 4067428375, 7200210523, 12693890803, 22290727268, 38993410516, 67959010130, 118016656268, 204233654229, 352245710866, 605538866862, 1037668522922, 1772700955975, 3019333854177, 5127694484375, 8683676638832, 14665233966068, 24700752691832, 41495176877972, 69531305679518
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| An ordinary partition is a row of numbers in non-decreasing order whose sum is n. Here the numbers are in a three-dimensional pile, non-decreasing in the x-, y- and z-directions.
Finding a g.f. for this sequence is an unsolved problem. At first it was thought that it was given by A000294.
Contribution from Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2009: Equals A000041 convolved with A002836: [1, 0, 2, 5, 12, 24, 56, 113,...] and row sums of the convolution triangle A161564.
|
|
|
REFERENCES
| A. O. L. Atkin, P. Bratley, I. G. McDonald and J. K. S. McKay, Some computations for m-dimensional partitions, Proc. Camb. Phil. Soc., 63 (1967), 1097-1100.
Srivatsan Balakrishnan, Suresh Govindarajan and Naveen S. Prabhakar, On the asymptotics of higher-dimensional partitions, arXiv:1105.6231.
P. Bratley and J. K. S. McKay, Algorithm 313: Multi-dimensional partition generator, Comm. ACM, 10 (Issue 10, 1967), p. 666.
D. E. Knuth, A note on solid partitions, Math. Comp., 24 (1970), 955-961.
P. A. MacMahon, Memoir on the theory of partitions of numbers - Part VI, Phil. Trans. Roal Soc., 211 (1912), 345-373.
P. A. MacMahon, Combinatory Analysis. Cambridge Univ. Press, London and New York, Vol. 1, 1915 and Vol. 2, 1916; see vol. 2, p 332.
S. P. Naveen, On The Asymptotics of Some Counting Problems in Physics, Thesis, Bachelor of Technology, DEPARTMENT OF PHYSICS, INDIAN INSTITUTE OF TECHNOLOGY, MADRAS, May 2011; http://www.physics.iitm.ac.in/~suresh/theses/NaveenThesis.pdf
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| Suresh Govindarajan, Table of n, a(n) for n = 0..68
Suresh Govindarajan, Solid Partitions Project Dec 14, 2010
P. A. MacMahon, Combinatory analysis.
Ville Mustonen and R. Rajesh, Numerical Estimation of the Asymptotic Behaviour of Solid Partitions of an Integer, J. Phys. A 36 (2003), no. 24, 6651-6659.
Eric Weisstein's World of Mathematics, Solid Partition
|
|
|
EXAMPLE
| Examples for n=2 and n=3.
a(2) = 4: 2; 11 where the first 1 is at the origin and the second 1 is in the x, y or z direction.
a(3) = 10: 3; 21 where the 2 is at the origin and the 1 is on the x, y or z axis; 111 (a row of 3 ones on the x, y or z axes); and three 1's with one 1 at the origin and the other two 1's on two of the three axes.
|
|
|
CROSSREFS
| Cf. A000041, A000219, A000294, A000334, A000390, A002835, A002836, A005980, A037452, A080207, A082535.
Cf. A002836, A000041, A161564 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 13 2009]
Sequence in context: A200455 A022812 A192306 * A000294 A133086 A178037
Adjacent sequences: A000290 A000291 A000292 * A000294 A000295 A000296
|
|
|
KEYWORD
| nonn,nice
|
|
|
AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
|
|
|
EXTENSIONS
| More terms from the Mustonen and Rajesh article, May 02 2003.
a(51)-a(62) found by Suresh Govindarajan and students, Dec 14 2010
a(63)-a(68) found by Suresh Govindarajan and students on June 1, 2011
|
| |
|
|
