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 A066186 Sum of all parts of all partitions of n. 137

%I

%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  --> ,  --> ,  --> ,  --> ,  --> ,  --> ,  --> 2,  --> 2,  --> 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  --> ,  --> ,  --> ,  --> . - _Thomas Wieder_, May 20 2004

%C Row sums of triangle A138785 and of triangle A181187. - _Omar E. Pol_, Feb 26 2012

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

%C Row sums of triangle A221529. - _Omar E. Pol_, Jan 21 2013

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

%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 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 [prod_{k>0} 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

%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 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

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

%K easy,nonn,nice

%O 0,3

%A _Wouter Meeussen_, Dec 15 2001