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%I M2575 #76 Nov 25 2020 05:07:05
%S 1,1,3,6,14,25,56,97,198,354,672,1170,2207,3762,6786,11675,20524,
%T 34636,60258,100580,171894,285820,480497,791316,1321346,2156830,
%U 3557353,5783660,9452658,15250216,24771526,39713788,64011924,102199026,163583054,259745051
%N a(n) is the sum of products of terms in all partitions of n.
%C 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 Same parity as A000009. - _Jon Perry_, Feb 12 2004
%D G. Labelle, personal communication.
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Alois P. Heinz, <a href="/A006906/b006906.txt">Table of n, a(n) for n = 0..6000</a> (first 1001 terms from T. D. Noe)
%H Atreya Chatterjee, <a href="https://arxiv.org/abs/2006.01170">Emergent gravity from patterns in natural numbers</a>, arXiv:2006.01170 [gr-qc], 2020.
%H Dean Hickerson, <a href="/A006906/a006906.txt">Comments on A006906</a>
%H Robert Schneider and Andrew V. Sills, <a href="http://math.colgate.edu/~integers/uproc13/uproc13.Abstract.html">The product of parts or 'norm' of a partition</a>, #A13 INTEGERS 20A (2020), Theorem 7, p. 4.
%F 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
%F 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
%F G.f.: 1 / Product_{k>=1} (1-k*x^k).
%F 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
%F a(n) = (1/n)*Sum_{k=1..n} A078308(k)*a(n-k). - _Vladeta Jovovic_, Nov 22 2002
%F O.g.f.: exp( Sum_{n>=1} Sum_{k>=1} k^n * x^(n*k) / n ). - _Paul D. Hanna_, Sep 18 2017
%F 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
%e Partitions of 0 are {()} whose products are {1} whose sum is 1.
%e Partitions of 1 are {(1)} whose products are {1} whose sum is 1.
%e Partitions of 2 are {(2),(1,1)} whose products are {2,1} whose sum is 3.
%e Partitions of 3 are 3 => {(3),(2,1),(1,1,1)} whose products are {3,2,1} whose sum is 6.
%e 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.
%p A006906 := proc(n)
%p option remember;
%p if n = 0 then
%p 1;
%p else
%p add( A078308(k)*procname(n-k),k=1..n)/n ;
%p end if;
%p end proc: # _R. J. Mathar_, Dec 14 2011
%p # second Maple program:
%p b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p b(n, i-1) +add(b(n-i*j, i-1)*(i^j), j=1..n/i)))
%p end:
%p a:= n-> b(n, n):
%p seq(a(n), n=0..40); # _Alois P. Heinz_, Feb 25 2013
%t (* 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 *)
%t Table[Total[Times@@@IntegerPartitions[n]],{n,0,35}] (* _Harvey P. Dale_, Jan 14 2013 *)
%t nmax = 40; CoefficientList[Series[Product[1/(1 - k*x^k), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 15 2015 *)
%t nmax = 40; CoefficientList[Series[Exp[Sum[PolyLog[-j, x^j]/j, {j, 1, nmax}]], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Dec 15 2015 *)
%o (Haskell)
%o a006906 n = p 1 n 1 where
%o p _ 0 s = s
%o p k m s | m<k = 0 | otherwise = p k (m-k) (k*s) + p (k+1) m s
%o -- _Reinhard Zumkeller_, Dec 07 2011
%Y Row sums of A118851.
%Y Cf. A000041, A007870, A022629, A022661, A022693, A077335, A163318, A265758, A302830, A318127, A322364, A322365.
%K nonn,nice,easy
%O 0,3
%A _Simon Plouffe_
%E More terms from _Vladeta Jovovic_, Oct 04 2001
%E Edited by _N. J. A. Sloane_, May 19 2007
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