cp's OEIS Frontend

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)

(Sum over row n) = A000041(n) = number of partitions of n.

Reversal of this array is array in A113686.

From Gary W. Adamson, Apr 11 2010: (Start)

Let M = an infinite lower triangular matrix with A000041 interleaved with zeros: (1, 0, 1, 0, 2, 0, 3, 0, 5, ...) and Q = A000009 diagonalized with the rest zeros.

Then A113685 = M*Q. That row sums of the triangle (deleting prefaced zeros) = A000041 is equivalent to the identity: p(x) = p(x^2) * A000009(x). (End)

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798, sum A056239(n).

Also sum of all odd-indexed parts minus the sum of all even-indexed parts of all partitions of n (Cf. A206563). - Omar E. Pol, Feb 12 2012

Column 1 of A206563. - Omar E. Pol, Feb 15 2012

Suppose that p=[p(1),p(2),p(3),...] is a partition of n with parts in nonincreasing order. Let f(p) = p(1) - p(2) + p(3) - ... be the alternating sum of parts of p and let F(n) = sum of alternating sums of all partitions of n. Conjecture: F(n) = A066897(n) for n >= 1. - Clark Kimberling, May 17 2019

From Omar E. Pol, Apr 02 2023: (Start)

Convolution of A000041 and A001227.

Convolution of A002865 and A060831.

a(n) is also the total number of odd 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 odd divisors are also all odd parts of all partitions of n. (End)

a(n) is odd iff n is a term of A067567 (proof: n*p(n) = the sum of the parts in all the partitions of n == the number of odd parts in all partitions of n (mod 2). Hence the number of odd parts in all partitions of n is odd iff n*p(n) is odd, equivalently, iff both n and p(n) are odd). - Peter Bala, Jan 11 2025

The version for all k = 0..n is A113685 (including zeros).

Row sums = A000041, the partition numbers.

The current triangle is the 2nd in an infinite set, followed by A174714 (k=3), and A174715, (k=4); in which row sums of each triangle = A000041.

k-th triangle in the infinite set can be defined as having the sequence:

"Euler transform of ones: (1,1,1,...) interleaved with (k-1) zeros"; shifted down k times (except column 0) in successive columns, then multiplied * triangle A174712, the diagonalized variant of A000041, A174713 begins with A000009 shifted down twice (triangle A173305); where A000009 = the Euler transform of period 2 sequence: [1,0,1,0,...].

Similarly, triangle A174714 begins with A000716 shifted down thrice; where A000716 = the Euler transform of period 3 series: [1,1,0,1,1,0,...]. Then multiply the latter as an infinite lower triangular matrix * A174712, the diagonalized variant of A000041, obtaining triangle A174714 with row sums = A000041.

Case k=4 = triangle A174715 which begins with the Euler transform of period 4 series: [1,1,1,0,1,1,1,0,...], shifted down 4 times in successive columns then multiplied * A174712, the diagonalized variant of A000041.

All triangles in the infinite set have row sums = A000041.

The sequences: "Euler transform of ones interleaved with (k-1) zeros" have the following properties, beginning with k=2:

...

k=2, A000009: = Euler transform of [1,0,1,0,1,0,...] and satisfies

.....A000009. = p(x)/p(x^2), where p(x) = polcoeff A000041; and A000041 =

.....A000009(x) = r(x), then p(x) = r(x) * r(x^2) * r(x^4) * r(x^8) * ...

...

k=3, A000726: = Euler transform of [1,1,0,1,1,0,...] and satisfies

.....A000726(x): = p(x)/p(x^3), and given s(x) = polcoeff A000726, we get

.....A000041(x) = p(x) = s(x) * s(x^3) * s(x^9) * s(x^27) * ...

...

k=4, A001935: = Euler transform of [1,1,1,0,1,1,1,0,...] and satisfies

.....A001935(x) = p(x)/p(x^4) and given t(x) = polcoeff A001935, we get

.....A000041(x) = p(x) = t(x) * t(x^4) * t(x^16) * t(x^64) * ...

...

Also the number of integer partitions of n whose even parts sum to k, for k an even number from zero to n. The version including odd k is A113686. - Gus Wiseman, Oct 23 2023

Partial sums of A206436. - Omar E. Pol, Mar 17 2012

From Omar E. Pol, Apr 02 2023: (Start)

Convolution of A000041 and A146076.

Convolution of A002865 and A271342.

a(n) is also the sum of all even 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 even divisors are also all even parts of all partitions of n. (End)

From Omar E. Pol, Apr 09 2023: (Start)

Convolution of A002865 and A000593.

a(n) is also the total sum of odd divisors of the terms in the n-th row of the triangle A336811.

a(n) is also the sum of odd terms in the n-th row of the triangle A207378.

a(n) is also the sum of odd terms in the n-th row of the triangle A336812. (End)

Essentially this sequence is related to A206561 in the same way as A206563 is related to A181187. See the calculation in the example section of A206563.