|
| |
|
|
A053445
|
|
Second differences of partition numbers A000041.
|
|
28
| |
|
|
1, 0, 1, 0, 2, 0, 3, 1, 4, 2, 7, 3, 10, 7, 14, 11, 22, 17, 32, 28, 45, 43, 67, 63, 95, 96, 134, 139, 192, 199, 269, 287, 373, 406, 521, 566, 718, 792, 983, 1092, 1346, 1496, 1827, 2045, 2465, 2772, 3323, 3733, 4449, 5016, 5929, 6696, 7882, 8897, 10426
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,5
|
|
|
COMMENTS
| First differences of 0 1 1 2 2 4 4 7 8 12 14 21 24 34 41 55... (A002865).
For n>2, a(n-2) is the number of partitions of n with all parts > 1 and with the largest part occurring more than once. The list of partitions counted begins 22 (so a(2) = 1); 33, 222 (so a(4) = 2); 44, 332, 2222 (so a(6) = 3); 333; 55, 442, 3322, 22222; 443, 3332; 66, 552, 444, 4422, 3333, 33222, 222222; 553, 4432, 33322; ...
a(n) is the number of certain level-n quasi-primary states of a quotient space of certain Verma modules. See the Furlan et al. reference p. 67. - Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Apr 25 2003
|
|
|
REFERENCES
| P. Furlan, G. M. Sotkov and I. T. Todorov, Two-Dimensional Conformal Quantum Field Theory, Rivista d. Nuovo Cimento 12, 6 (1989) 1-202.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 115, 3.
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
Almkvist, Gert, "On the differences of the partition function", Acta Arith., 61.2 (1992), 173-181.
|
|
|
EXAMPLE
| a(8) = 7 - 4 = 3; the corresponding partitions are 44, 332 and 2222
|
|
|
MATHEMATICA
| Table[(PartitionsP[n+2]-PartitionsP[n+1])-(PartitionsP[n+1]-PartitionsP[n]), {n, 0, 42}] - Vladimir Orlovsky (4vladimir(AT)gmail.com), Apr 23 2008
Differences[Table[ PartitionsP[n], {n, 0, 56}], 2] (* From Jean-François Alcover, Sep 07 2011 *)
|
|
|
PROG
| (MAGMA) m:=58; S:=[ NumberOfPartitions(n): n in [0..m] ]; [ S[n+2]-2*S[n+1]+S[n]: n in [1..m-2] ]; [From Klaus Brockhaus (klaus-brockhaus(AT)t-online.de), Jun 09 2009]
|
|
|
CROSSREFS
| Cf. A000041, A002865, A072380, A081094, A081095.
Sequence in context: A084964 A008720 A008734 * A175990 A162517 A180876
Adjacent sequences: A053442 A053443 A053444 * A053446 A053447 A053448
|
|
|
KEYWORD
| easy,nice,nonn
|
|
|
AUTHOR
| Alford Arnold (Alford1940(AT)aol.com), Jan 12 2000
|
|
|
EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Feb 02 2000
Start of sequence changed Apr 25 2003
|
| |
|
|
