|
Search: id:A000097
|
|
|
| A000097 |
|
Number of partitions of n if there are two kinds of 1's and two kinds of 2's.
(Formerly M1361 N0525)
|
|
+0
12
|
|
| 1, 2, 5, 9, 17, 28, 47, 73, 114, 170, 253, 365, 525, 738, 1033, 1422, 1948, 2634, 3545, 4721, 6259, 8227, 10767, 13990, 18105, 23286, 29837, 38028, 48297, 61053, 76926, 96524, 120746, 150487, 187019, 231643, 286152, 352413, 432937, 530383, 648245
(list; graph; listen)
|
|
|
OFFSET
|
0,2
|
|
|
COMMENT
|
Also number of partitions of 2*n with exactly 2 odd parts (offset 1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 12 2005
Also number of transitions from one partition of n+2 to another, where a transition consists of replacing any two parts with their sum. Remove all 1' and 2' from the partition, replacing them with ((number of 2') + 1) and ((number of 1') + (number of 2') + 1); these are the two parts being summed. Number of partitions of n into parts of 2 kinds with at most 2 parts of the second kind, or of n+2 into parts of 2 kinds with exactly 2 parts of the second kind. - Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 20 2006
|
|
REFERENCES
|
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, Vol. 4, Cambridge Univ. Press, 1958, p. 90.
J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
|
|
LINKS
|
T. D. Noe, Table of n, a(n) for n=0..1000
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
N. J. A. Sloane, Transforms
|
|
FORMULA
|
Euler transform of 2 2 1 1 1 1 1...
G.f.=1/[(1-x)(1-x^2)*product((1-x^k), k=1..infinity)].
a(n)=sum(A000070(n-2*j), j=0..floor(n/2)), n>=0.
|
|
EXAMPLE
|
a(3)=9 because we have 3, 2+1, 2+1', 2'+1, 2'+1', 1+1+1, 1+1+1', 1+1'+1' and 1'+1'+1'.
|
|
MAPLE
|
with (numtheory): etr:= proc(p) local b; b:=proc(n) option remember; local d, j; if n=0 then 1 else add (add (d*p(d), d=divisors(j)) *b(n-j), j=1..n)/n fi end end: a:= etr (n->`if`(n<3, 2, 1)): seq (a(n), n=0..40); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 08 2008]
|
|
CROSSREFS
|
First differences are in A024786.
Cf. A000070, A008951, A000098, A000710.
Third column of Riordan triangle A008951 and of triangle A103923.
Sequence in context: A139672 A093694 A068006 this_sequence A081996 A034329 A133470
Adjacent sequences: A000094 A000095 A000096 this_sequence A000098 A000099 A000100
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
N. J. A. Sloane (njas(AT)research.att.com).
|
|
EXTENSIONS
|
More terms from Pab Ter (pabrlos(AT)yahoo.com), May 04 2004
Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 23 2005
More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 20 2006
|
|
|
Search completed in 0.003 seconds
|
