|
| |
|
|
A006906
|
|
a(n) is the sum of products of terms in all partitions of n.
(Formerly M2575)
|
|
73
|
|
|
|
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
|
Alois P. Heinz, Table of n, a(n) for n = 0..6000 (first 1001 terms from T. D. Noe)
Atreya Chatterjee, Emergent gravity from patterns in natural numbers, arXiv:2006.01170 [gr-qc], 2020.
Dean Hickerson, Comments on A006906
Robert Schneider and Andrew V. Sills, The product of parts or 'norm' of a partition, #A13 INTEGERS 20A (2020), Theorem 7, p. 4.
|
|
|
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. (See the Links entry.) - Dean Hickerson, Aug 19 2007
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_{k>=1} (1-k*x^k).
G.f.: 1 + Sum_{n>=1} n*x^n / Product_{k=1..n} (1-k*x^k) = 1 + Sum_{n>=1} n*x^n / Product_{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
|
|
|
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
|
Row sums of A118851.
Cf. A000041, A007870, A022629, A022661, A022693, A077335, A163318, A265758, A302830, A318127, A322364, A322365.
Sequence in context: A236429 A316245 A002219 * A324703 A120940 A049940
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
|
| |
|
|
