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A000097
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Number of partitions of n if there are two kinds of 1's and two kinds of 2's.
(Formerly M1361 N0525)
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12
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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; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| 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
Contribution from Christian Gutschwager (gutschwager(AT)math.uni-hannover.de), Feb 10 2010: (Start)
a(n) is also the number of pairs of partitions of n+2 which differ by only one box (for bijection see Gutschwager link).
a(n) is also the number of partitions of n with two parts marked
a(n) is also the number of partitions of n+1 with two different parts marked (End)
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REFERENCES
| 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.
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).
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LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
Christian Gutschwager, Skew characters which contain only few components [From Christian Gutschwager (gutschwager(AT)math.uni-hannover.de), Feb 10 2010]
P. J. Cameron, Sequences realized by oligomorphic permutation groups, J. Integ. Seqs. Vol. 3 (2000), #00.1.5.
N. J. A. Sloane, Transforms
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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.
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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'.
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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]
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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 * A081996 A034329 A133470
Adjacent sequences: A000094 A000095 A000096 * A000098 A000099 A000100
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KEYWORD
| nonn,easy
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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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
Edited by Charles R Greathouse IV (charles.greathouse(AT)case.edu), Apr 20 2010
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