

A066186


Sum of all parts of all partitions of n.


182



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
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OFFSET

0,3


COMMENTS

Sum of the zeroth moments of all partitions of n.
Also the number of oneelement transitions from the integer partitions of n to the partitions of n1 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 oneelement transitions from the integer partitions of n to the partitions of n1 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
Apart from initial zero this is also as follows:
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(n1) respectively. The polycube looks like a symmetric tower (cf. A221529). A dissection is a threedimensional spiral whose top view is described in A239660. The growth of the volume of the polycube represents each convolution mentioned above. (End)
a(n) is also the sum of all divisors of all positive integers in a sequence with n blocks where the mth block consists of A000041(nm) 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)


LINKS



FORMULA

G.f.: x * (d/dx) Product_{k>=1} 1/(1x^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)
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


EXAMPLE

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.


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]


PROG

(Haskell)
a066186 = sum . concat . ps 1 where
ps _ 0 = [[]]
ps i j = [t:ts  t < [i..j], ts < ps t (j  t)]
(Sage)
[n*Partitions(n).cardinality() for n in range(41)] # Peter Luschny, Jul 29 2014
(Python)
from sympy import npartitions


CROSSREFS

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


KEYWORD

easy,nonn,nice


AUTHOR



EXTENSIONS



STATUS

approved



