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A006906 a(n) = sum of products of terms in all partitions of n.
(Formerly M2575)
36
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 (list; graph; refs; listen; history; text; 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, 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, 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

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

# 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

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, 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 *)

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. 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

KEYWORD

nonn,nice,easy

AUTHOR

Simon Plouffe

EXTENSIONS

More terms from Vladeta Jovovic, Oct 04 2001

Edited by N. J. A. Sloane, May 19 2007

STATUS

approved

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Last modified June 25 20:26 EDT 2017. Contains 288730 sequences.