|
| |
|
|
A006906
|
|
a(n) = sum of products of terms in all partitions of n.
(Formerly M2575)
|
|
15
| |
|
|
1, 1, 3, 6, 14, 25, 56, 97, 198, 354, 672, 1170, 2207, 3762, 6786, 11675, 20524, 34636, 60258, 100580, 171894, 285820, 480497, 791316, 1321346, 2156830, 3557353, 5783660, 9452658, 15250216, 24771526, 39713788, 64011924, 102199026
(list; graph; refs; listen; history; internal format)
|
|
|
|
OFFSET
| 0,3
|
|
|
COMMENTS
| a(0) = 1 since the only partition of 0 is the empty partition. The product of its terms is the empty product, namely 1.
Same parity as A000009. - Jon Perry (perry(AT)globalnet.co.uk), Feb 12 2004
|
|
|
REFERENCES
| G. Labelle, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
|
|
|
LINKS
| T. D. Noe, Table of n, a(n) for n=0..1000
Dean Hickerson, Comments on A006906
|
|
|
FORMULA
| 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.
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
G.f.: 1 / Product (1-k*x^k).
G.f.: 1 + sum(n>=1, n*x^n / prod(k=1..n, 1-k*x^k) ) = 1 + sum(n>=1, n*x^n / prod(k>=n, 1-k*x^k) ). [Joerg Arndt, Mar 23 2011]
a(n) = (1/n)*Sum_{k=1..n} A078308(k)*a(n-k). - Vladeta Jovovic (vladeta(AT)eunet.rs), Nov 22 2002
|
|
|
EXAMPLE
| Partitions of 0 are {()} whose products are {1} whose sum is 1
Partitions of 1 are {(1)} whose products are {1} whose sum is 1
Partitions of 2 are {(2),(1,1)} whose products are {2,1} whose sum is 3
Partitions of 3 are 3 => {(3),(2,1),(1,1,1)} whose products are {3,2,1} whose sum is 6
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
|
|
|
MAPLE
| A006906 := proc(n)
option remember;
if n = 0 then
1;
else
add( A078308(k)*procname(n-k), k=1..n)/n ;
end if;
end proc: # R. J. Mathar, Dec 14 2011
|
|
|
MATHEMATICA
| (* 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
|
|
|
PROG
| (Haskell)
a006906 n = p 1 n 1 where
p _ 0 s = s
p k m s | m<k = 0 | otherwise = p k (m-k) (k*s) + p (k+1) m s
-- Reinhard Zumkeller, Dec 07 2011
|
|
|
CROSSREFS
| Cf. A007870.
Cf. A000041.
Sequence in context: A026271 A166212 A002219 * A120940 A049940 A051749
Adjacent sequences: A006903 A006904 A006905 * A006907 A006908 A006909
|
|
|
KEYWORD
| nonn,nice,easy
|
|
|
AUTHOR
| Simon Plouffe (simon.plouffe(AT)gmail.com)
|
|
|
EXTENSIONS
| More terms from Vladeta Jovovic (vladeta(AT)eunet.rs), Oct 04 2001
Edited by N. J. A. Sloane (njas(AT)research.att.com), May 19 2007
|
| |
|
|
