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A006906
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a(n) = sum of products of terms in all partitions of n.
(Formerly M2575)
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37
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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, 163583054, 259745051
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OFFSET
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0,3
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COMMENTS
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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, Feb 12 2004
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REFERENCES
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G. Labelle, personal communication.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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Alois P. Heinz, Table of n, a(n) for n = 0..6000 (first 1001 terms from T. D. Noe)
Dean Hickerson, Comments on A006906
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FORMULA
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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, 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, 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, Nov 22 2002
O.g.f.: exp( Sum_{n>=1} Sum_{k>=1} k^n * x^(n*k) / n ). - Paul D. Hanna, Sep 18 2017
O.g.f.: exp( Sum_{n>=1} Sum_{k=1..n} A008292(n,k)*x^(n*k)/(n*(1-x^n)^(n+1)) ), where A008292 is the Eulerian numbers. - Paul D. Hanna, Sep 18 2017
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EXAMPLE
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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
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MAPLE
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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
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
b(n, i-1) +add(b(n-i*j, i-1)*(i^j), j=1..n/i)))
end:
a:= n-> b(n, n):
seq(a(n), n=0..40); # Alois P. Heinz, Feb 25 2013
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MATHEMATICA
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(* 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, Aug 19 2007
Table[Total[Times@@@IntegerPartitions[n]], {n, 0, 35}] (* Harvey P. Dale, Jan 14 2013 *)
nmax = 40; CoefficientList[Series[Product[1/(1 - k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)
nmax = 40; CoefficientList[Series[Exp[Sum[PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* Vaclav Kotesovec, Dec 15 2015 *)
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PROG
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(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
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CROSSREFS
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Cf. A000041, A007870, A022629, A022661, A022693, A077335, A163318, A265758.
Sequence in context: A285460 A236429 A002219 * A120940 A049940 A265947
Adjacent sequences: A006903 A006904 A006905 * A006907 A006908 A006909
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KEYWORD
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nonn,nice,easy,changed
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AUTHOR
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Simon Plouffe
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EXTENSIONS
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More terms from Vladeta Jovovic, Oct 04 2001
Edited by N. J. A. Sloane, May 19 2007
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STATUS
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approved
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