login
The OEIS is supported by the many generous donors to the OEIS Foundation.

 

Logo
Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A066186 Sum of all parts of all partitions of n. 182

%I #212 Oct 22 2023 18:21:31

%S 0,1,4,9,20,35,66,105,176,270,420,616,924,1313,1890,2640,3696,5049,

%T 6930,9310,12540,16632,22044,28865,37800,48950,63336,81270,104104,

%U 132385,168120,212102,267168,334719,418540,520905,647172,800569,988570,1216215,1493520

%N Sum of all parts of all partitions of n.

%C Sum of the zeroth moments of all partitions of n.

%C Also the number of one-element transitions from the integer partitions of n to the partitions of n-1 for labeled parts with the assumption that any part z is composed of labeled elements of amount 1, i.e., z = 1_1 + 1_2 + ... + 1_z. Then one can take from z a single element in z different ways. E.g., for n=3 to n=2 we have A066186(3) = 9 and [111] --> [11], [111] --> [11], [111] --> [11], [12] --> [111], [12] --> [111], [12] --> [2], [3] --> 2, [3] --> 2, [3] --> 2. For the unlabeled case, one can take a single element from z in only one way. Then the number of one-element transitions from the integer partitions of n to the partitions of n-1 is given by A000070. E.g., A000070(3) = 4 and for the transition from n=3 to n=2 one has [111] --> [11], [12] --> [11], [12] --> [2], [3] --> [2]. - _Thomas Wieder_, May 20 2004

%C Also sum of all parts of all regions of n (Cf. A206437). - _Omar E. Pol_, Jan 13 2013

%C From _Omar E. Pol_, Jan 19 2021: (Start)

%C Apart from initial zero this is also as follows:

%C Convolution of A000203 and A000041.

%C Convolution of A024916 and A002865.

%C For n >= 1, a(n) is also the number of cells in a symmetric polycube in which the terraces are the symmetric representation of sigma(k), for k = n..1, (cf. A237593) starting from the base and located at the levels A000041(0)..A000041(n-1) respectively. The polycube looks like a symmetric tower (cf. A221529). A dissection is a three-dimensional spiral whose top view is described in A239660. The growth of the volume of the polycube represents each convolution mentioned above. (End)

%C From _Omar E. Pol_, Feb 04 2021: (Start)

%C a(n) is also the sum of all divisors of all positive integers in a sequence with n blocks where the m-th block consists of A000041(n-m) copies of m, with 1 <= m <= n. The mentioned divisors are also all parts of all partitions of n.

%C Apart from initial zero this is also the convolution of A340793 and A000070. (End)

%H Vincenzo Librandi, <a href="/A066186/b066186.txt">Table of n, a(n) for n = 0..1000</a>

%H F. G. Garvan, <a href="http://dx.doi.org/10.1016/j.aim.2011.05.013">Higher-order spt functions</a>, Adv. Math. 228 (2011), no. 1, 241-265, <a href="http://qseries.org/fgarvan/papers/hspt.pdf">alternate copy</a>. - From _N. J. A. Sloane_, Jan 02 2013

%H F. G. Garvan, <a href="http://arxiv.org/abs/1008.1207">Higher-order spt functions</a>, arXiv:1008.1207 [math.NT], 2010.

%H T. J. Osler, A. Hassen and T. R. Chandrupatia, <a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.538.6485">Surprising connections between partitions and divisors</a>, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).

%H Omar E. Pol, <a href="/A066186/a066186_1.jpg">Illustration of a(10), prism and tower</a>, each polycube contains 420 cubes.

%H Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa404.jpg">Illustration of initial terms of A066186 and of A139582 (n>=1)</a>

%F a(n) = n * A000041(n). - _Omar E. Pol_, Oct 10 2011

%F G.f.: x * (d/dx) Product_{k>=1} 1/(1-x^k), i.e., derivative of g.f. for A000041. - _Jon Perry_, Mar 17 2004 (adjusted to match the offset by _Geoffrey Critzer_, Nov 29 2014)

%F Equals A132825 * [1, 2, 3, ...]. - _Gary W. Adamson_, Sep 02 2007

%F a(n) = A066967(n) + A066966(n). - _Omar E. Pol_, Mar 10 2012

%F a(n) = A207381(n) + A207382(n). - _Omar E. Pol_, Mar 13 2012

%F a(n) = A006128(n) + A196087(n). - _Omar E. Pol_, Apr 22 2012

%F a(n) = A220909(n)/2. - _Omar E. Pol_, Jan 13 2013

%F a(n) = Sum_{k=1..n} A000203(k)*A000041(n-k), n >= 1. - _Omar E. Pol_, Jan 20 2013

%F a(n) = Sum_{k=1..n} k*A036043(n,n-k+1). - _L. Edson Jeffery_, Aug 03 2013

%F a(n) = Sum_{k=1..n} A024916(k)*A002865(n-k), n >= 1. - _Omar E. Pol_, Jul 13 2014

%F a(n) ~ exp(Pi*sqrt(2*n/3))/(4*sqrt(3)) * (1 - (sqrt(3/2)/Pi + Pi/(24*sqrt(6))) / sqrt(n)). - _Vaclav Kotesovec_, Oct 24 2016

%F a(n) = Sum_{k=1..n} A340793(k)*A000070(n-k), n >= 1. - _Omar E. Pol_, Feb 04 2021

%e a(3)=9 because the partitions of 3 are: 3, 2+1 and 1+1+1; and (3) + (2+1) + (1+1+1) = 9.

%e a(4)=20 because A000041(4)=5 and 4*5=20.

%p with(combinat): a:= n-> n*numbpart(n): seq(a(n), n=0..50); # _Zerinvary Lajos_, Apr 25 2007

%t PartitionsP[ Range[0, 60] ] * Range[0, 60]

%o (PARI) a(n)=numbpart(n)*n \\ _Charles R Greathouse IV_, Mar 10 2012

%o (Haskell)

%o a066186 = sum . concat . ps 1 where

%o ps _ 0 = [[]]

%o ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]

%o -- _Reinhard Zumkeller_, Jul 13 2013

%o (Sage)

%o [n*Partitions(n).cardinality() for n in range(41)] # _Peter Luschny_, Jul 29 2014

%o (Python)

%o from sympy import npartitions

%o def A066186(n): return n*npartitions(n) # _Chai Wah Wu_, Oct 22 2023

%Y Cf. A000041, A093694, A000070, A132825, A001787 (same for ordered partitions), A277029, A000203, A221529, A237593, A239660.

%Y First differences give A138879. - _Omar E. Pol_, Aug 16 2013

%Y Row sums of triangles A138785, A181187, A245099, A337209, A339106, A340423, A340424, A221529, A302246, A338156, A340035, A340056, A340057, A346741. - _Omar E. Pol_, Aug 02 2021

%K easy,nonn,nice

%O 0,3

%A _Wouter Meeussen_, Dec 15 2001

%E a(0) added by _Franklin T. Adams-Watters_, Jul 28 2014

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recents
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified March 28 10:55 EDT 2024. Contains 371241 sequences. (Running on oeis4.)