A066186 Sum of all parts of all partitions of n.

0, 1, 4, 9, 20, 35, 66, 105, 176, 270, 420, 616, 924, 1313, 1890, 2640, 3696, 5049, 6930, 9310, 12540, 16632, 22044, 28865, 37800, 48950, 63336, 81270, 104104, 132385, 168120, 212102, 267168, 334719, 418540, 520905, 647172, 800569, 988570, 1216215, 1493520

Offset: 0

Author

Wouter Meeussen, Dec 15 2001

Keywords

• easy
• nonn
• nice

Comments

Sum of the zeroth moments of all partitions of n.

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

Also sum of all parts of all regions of n (Cf. A206437). - Omar E. Pol, Jan 13 2013

From Omar E. Pol, Jan 19 2021: (Start)

Apart from initial zero this is also as follows:

Convolution of A000203 and A000041.

Convolution of A024916 and A002865.

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)

From Omar E. Pol, Feb 04 2021: (Start)

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.

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

Examples

			a(3)=9 because the partitions of 3 are: 3, 2+1 and 1+1+1; and (3) + (2+1) + (1+1+1) = 9.
a(4)=20 because A000041(4)=5 and 4*5=20.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• F. G. Garvan, Higher-order spt functions, Adv. Math. 228 (2011), no. 1, 241-265, alternate copy. - From _N. J. A. Sloane_, Jan 02 2013
• F. G. Garvan, Higher-order spt functions, arXiv:1008.1207 [math.NT], 2010.
• T. J. Osler, A. Hassen and T. R. Chandrupatia, Surprising connections between partitions and divisors, The College Mathematics Journal, Vol. 38. No. 4, Sep. 2007, 278-287 (see p. 287).
• Omar E. Pol, Illustration of a(10), prism and tower, each polycube contains 420 cubes.
• Omar E. Pol, Illustration of initial terms of A066186 and of A139582 (n>=1)

Crossrefs

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

First differences give A138879. - Omar E. Pol, Aug 16 2013

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

Programs

Haskell

a066186 = sum . concat . ps 1 where
   ps _ 0 = [[]]
   ps i j = [t:ts | t <- [i..j], ts <- ps t (j - t)]
-- Reinhard Zumkeller, Jul 13 2013

Maple

with(combinat): a:= n-> n*numbpart(n): seq(a(n), n=0..50); # Zerinvary Lajos, Apr 25 2007

Mathematica

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

PARI

a(n)=numbpart(n)*n \\ Charles R Greathouse IV, Mar 10 2012

Python

from sympy import npartitions
def A066186(n): return n*npartitions(n) # Chai Wah Wu, Oct 22 2023

Sage

[n*Partitions(n).cardinality() for n in range(41)] # Peter Luschny, Jul 29 2014

Formula

a(n) = n * A000041(n). - Omar E. Pol, Oct 10 2011

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)

Equals A132825 * [1, 2, 3, ...]. - Gary W. Adamson, Sep 02 2007

a(n) = A066967(n) + A066966(n). - Omar E. Pol, Mar 10 2012

a(n) = A207381(n) + A207382(n). - Omar E. Pol, Mar 13 2012

a(n) = A006128(n) + A196087(n). - Omar E. Pol, Apr 22 2012

a(n) = A220909(n)/2. - Omar E. Pol, Jan 13 2013

a(n) = Sum_{k=1..n} A000203(k)*A000041(n-k), n >= 1. - Omar E. Pol, Jan 20 2013

a(n) = Sum_{k=1..n} k*A036043(n,n-k+1). - L. Edson Jeffery, Aug 03 2013

a(n) = Sum_{k=1..n} A024916(k)*A002865(n-k), n >= 1. - Omar E. Pol, Jul 13 2014

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

a(n) = Sum_{k=1..n} A340793(k)*A000070(n-k), n >= 1. - Omar E. Pol, Feb 04 2021

Extensions

a(0) added by Franklin T. Adams-Watters, Jul 28 2014