A277029 -id:A277029 - cp's OEIS frontend

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

Wouter Meeussen, Dec 15 2001

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)

			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.

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)

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

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

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

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

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

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

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

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

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

A340793 Sequence whose partial sums give A000203.

Original entry on oeis.org

1, 2, 1, 3, -1, 6, -4, 7, -2, 5, -6, 16, -14, 10, 0, 7, -13, 21, -19, 22, -10, 4, -12, 36, -29, 11, -2, 16, -26, 42, -40, 31, -15, 6, -6, 43, -53, 22, -4, 34, -48, 54, -52, 40, -6, -6, -24, 76, -67, 36, -21, 26, -44, 66, -48, 48, -40, 10, -30, 108, -106, 34, 8

Offset: 1

Omar E. Pol, Jan 21 2021

Essentially a duplicate of A053222.
Convolved with the nonzero terms of A000217 gives A175254, the volume of the stepped pyramid described in A245092.
Convolved with the nonzero terms of A046092 gives A244050, the volume of the stepped pyramid described in A244050.
Convolved with A000027 gives A024916.
Convolved with A000041 gives A138879.
Convolved with A000070 gives the nonzero terms of A066186.
Convolved with the nonzero terms of A002088 gives A086733.
Convolved with A014153 gives A182738.
Convolved with A024916 gives A000385.
Convolved with A036469 gives the nonzero terms of A277029.
Convolved with A091360 gives A276432.
Convolved with A143128 gives the nonzero terms of A000441.
For the correspondence between divisors and partitions see A336811.

1 together with A053222.
Cf. A000203 (partial sums).
Cf. A000027, A000041, A000070, A000217, A000385, A000441, A002088, A002961, A014153, A024916, A036469, A046092, A053223, A053224, A053225, A053226, A053227, A066186, A086733, A091360, A138879, A143128, A175254, A182738, A237593, A244050, A245092, A276432, A277029, A336811.

a:= n-> (s-> s(n)-s(n-1))(numtheory[sigma]):
seq(a(n), n=1..77);  # Alois P. Heinz, Jan 21 2021

Join[{1}, Differences @ Table[DivisorSigma[1, n], {n, 1, 100}]] (* Amiram Eldar, Jan 21 2021 *)

a(n) = if (n==1, 1, sigma(n)-sigma(n-1)); \\ Michel Marcus, Jan 22 2021

a(n) = A053222(n-1) for n>1. - Michel Marcus, Jan 22 2021

A276432 Sum of the traces of all plane partitions of n.

Original entry on oeis.org

1, 4, 10, 26, 56, 126, 252, 512, 980, 1866, 3427, 6258, 11121, 19618, 33975, 58328, 98732, 165804, 275246, 453544, 740338, 1200088, 1929897, 3083898, 4893775, 7720826, 12106814, 18883104, 29291740, 45215386, 69451631, 106197524, 161656759, 245050410, 369935066

Offset: 1

Emeric Deutsch, Sep 24 2016

nonn

Convolution of A000203 and A000219. - Vaclav Kotesovec, Sep 25 2016

Convolution of A340793 and A091360. - Omar E. Pol, Feb 16 2021

			a(3) = 10 because the 6 (=A000219(3)) planar partitions of 3 are [3], [2,1], [2;1], [1,1,1], [1;1;1], [1,1;1] (; indicates a new row); the sum of their traces is 3+2+2+1+1+1 = 10.

G. E. Andrews, The Theory of Partitions, Addison-Wesley, 1976, pp. 179-201.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000

Cf. A000219, A089353, A277029.

Cf. A091360, A340793.

g:= (sum(j*x^j/(1-x^j),j = 1..100))/(product((1-x^k)^k,k = 1..100)): gser := series(g, x = 0,40): seq(coeff(gser, x, m), m = 1 .. 35);
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, [1, 0], `if`(i<1, 0, add((p
      ->p+[0, j*p[1]])(b(n-i*j, i-1))*binomial(i+j-1, j), j=0..n/i)))
    end:
a:= n-> b(n$2)[2]:
seq(a(n), n=1..50);  # Alois P. Heinz, Sep 24 2018

nmax = 50; Rest[CoefficientList[Series[Sum[j*x^j/(1-x^j), {j, 1, nmax}]*Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Sep 25 2016 *)

G.f.: g(x) = Sum_{j>=1} (j*x^j/(1-x^j))/Product_{k>=1} (1-x^k)^k.

a(n) = Sum(k*A089353(n,k), k>=1).

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A066186 Sum of all parts of all partitions of n.

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A340793 Sequence whose partial sums give A000203.

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A276432 Sum of the traces of all plane partitions of n.

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