Search: id:A006906
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%I A006906 M2575
%S A006906 1,1,3,6,14,25,56,97,198,354,672,1170,2207,3762,6786,11675,20524,34636,
%T A006906 60258,100580,171894,285820,480497,791316,1321346,2156830,3557353,
%U A006906 5783660,9452658,15250216,24771526,39713788,64011924,102199026
%N A006906 a(n) = sum of products of terms in all partitions of n.
%C A006906 a(0) = 1 since the only partition of 0 is the empty partition. The product 
               of its terms is the empty product, namely 1.
%C A006906 Same parity as A000009. - Jon Perry (perry(AT)globalnet.co.uk), Feb 12 
               2004
%D A006906 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, 
               Academic Press, 1995 (includes this sequence).
%D A006906 G. Labelle, personal communication.
%H A006906 T. D. Noe, <a href="b006906.txt">Table of n, a(n) for n=0..1000</a>
%H A006906 Dean Hickerson, <a href="a006906.txt">Comments on A006906</a>
%F A006906 The limit of a(n+3)/a(n) is 3. However, the limit of a(n+1)/a(n) does 
               not exist. In fact, the sequence {a(n+1)/a(n)} has three limit points, 
               which are about 1.4422447, 1.4422491 and 1.4422549. - Dean Hickerson 
               (dean.hickerson(AT)yahoo.com), Aug 19 2007. See the link below.
%F A006906 a(n) ~ c(n mod 3) 3^(n/3), where c(0)=97923.26765718877..., c(1)=97922.93936857030... 
               and c(2)=97922.90546334208... - Dean Hickerson (dean.hickerson(AT)yahoo.com), 
               Aug 19 2007
%F A006906 G.f.: 1 / Product (1-kx^k).
%F A006906 a(n) = (1/n)*Sum_{k=1..n} A078308(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               Nov 22 2002
%e A006906 Partitions of 0 are {()} whose products are {1} whose sum is 1
%e A006906 Partitions of 1 are {(1)} whose products are {1} whose sum is 1
%e A006906 Partitions of 2 are {(2),(1,1)} whose products are {2,1} whose sum is 
               3
%e A006906 Partitions of 3 are 3 => {(3),(2,1),(1,1,1)} whose products are {3,2,
               1} whose sum is 6
%e A006906 Partitions of 4 are {(4),(3,1),(2,2),(2,1,1),(1,1,1,1)} whose products 
               are {4,3,4,2,1} whose sum is 14
%t A006906 (* a[n,k]=sum of products of partitions of n into parts <= k *) a[0,0]=1; 
               a[n_,0]:=0; a[n_,k_]:=If[k>n, a[n,n], a[n,k] = a[n,k-1] + k a[n-k,
               k] ]; a[n_]:=a[n,n] - Dean Hickerson (dean.hickerson(AT)yahoo.com), 
               Aug 19 2007
%Y A006906 Cf. A007870.
%Y A006906 Sequence in context: A026271 A166212 A002219 this_sequence A120940 A049940 
               A051749
%Y A006906 Adjacent sequences: A006903 A006904 A006905 this_sequence A006907 A006908 
               A006909
%K A006906 nonn,nice,easy
%O A006906 0,3
%A A006906 Simon Plouffe (simon.plouffe(AT)gmail.com)
%E A006906 More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 04 2001
%E A006906 Edited by N. J. A. Sloane (njas(AT)research.att.com), May 19 2007

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