Search: id:A000097
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%I A000097 M1361 N0525
%S A000097 1,2,5,9,17,28,47,73,114,170,253,365,525,738,1033,1422,1948,2634,3545,
%T A000097 4721,6259,8227,10767,13990,18105,23286,29837,38028,48297,61053,76926,
%U A000097 96524,120746,150487,187019,231643,286152,352413,432937,530383,648245
%N A000097 Number of partitions of n if there are two kinds of 1's and two kinds 
               of 2's.
%C A000097 Also number of partitions of 2*n with exactly 2 odd parts (offset 1). 
               - Vladeta Jovovic (vladeta(AT)eunet.rs), Jan 12 2005
%C A000097 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
%D A000097 H. Gupta et al., Tables of Partitions. Royal Society Mathematical Tables, 
               Vol. 4, Cambridge Univ. Press, 1958, p. 90.
%D A000097 J. Riordan, Combinatorial Identities, Wiley, 1968, p. 199.
%D A000097 N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 
               (includes this sequence).
%D A000097 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%H A000097 T. D. Noe, <a href="b000097.txt">Table of n, a(n) for n=0..1000</a>
%H A000097 P. J. Cameron, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               Sequences realized by oligomorphic permutation groups</a>, J. Integ. 
               Seqs. Vol. 3 (2000), #00.1.5.
%H A000097 N. J. A. Sloane, <a href="transforms.txt">Transforms</a>
%F A000097 Euler transform of 2 2 1 1 1 1 1...
%F A000097 G.f.=1/[(1-x)(1-x^2)*product((1-x^k), k=1..infinity)].
%F A000097 a(n)=sum(A000070(n-2*j), j=0..floor(n/2)), n>=0.
%e A000097 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'.
%p A000097 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]
%Y A000097 First differences are in A024786.
%Y A000097 Cf. A000070, A008951, A000098, A000710.
%Y A000097 Third column of Riordan triangle A008951 and of triangle A103923.
%Y A000097 Sequence in context: A139672 A093694 A068006 this_sequence A081996 A034329 
               A133470
%Y A000097 Adjacent sequences: A000094 A000095 A000096 this_sequence A000098 A000099 
               A000100
%K A000097 nonn,easy
%O A000097 0,2
%A A000097 N. J. A. Sloane (njas(AT)research.att.com).
%E A000097 More terms from Pab Ter (pabrlos(AT)yahoo.com), May 04 2004
%E A000097 Edited by Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 23 2005
%E A000097 More terms from Frank Adams-Watters (FrankTAW(AT)Netscape.net), Mar 20 
               2006

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