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In other words, a(n) is the number of graphic sequences of length n, where a graphic sequence is a sequence of numbers which can be the degree sequence of some graph.

In the article by A. Iványi, G. Gombos, L. Lucz, and T. Matuszka, "Parallel enumeration of degree sequences of simple graphs II", in Table 4 on page 260 the values a(30) = 7898574056034638 and a(31) = 30873429530206738 are incorrect due to the incorrect Gz(30) = 5876236938019300 and Gz(31) = 22974847474172100. - Wang Kai, Jun 05 2016

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. See A339658 for the Heinz numbers, and A339655 for the complement.

The following are equivalent characteristics for any positive integer n:

(1) the multiset of prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;

(2) n can be factored into distinct semiprimes;

(3) the unordered prime signature of n is loop-graphical.

I call loops of degree one half-loops, so these are half-loop-graphs or graphs with half-loops. - Gus Wiseman, Dec 31 2020

Equals the image of A181819 applied to the set of terms of A320892.

The Heinz number of an integer partition (y_1,...,y_k) is prime(y_1)*...*prime(y_k). This gives a bijective correspondence between positive integers and integer partitions.

An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices. Loop-graphical partitions are counted by A339656, with Heinz numbers A339658.

The following are equivalent characteristics for any positive integer n:

(1) the prime factors of n can be partitioned into distinct pairs, i.e., into a set of edges and loops;

(2) n can be factored into distinct semiprimes;

(3) the prime signature of n is loop-graphical.

A002494 is the number of graphs on n nodes with no isolated points and A095268 is the number of these graphs having distinct degree sequences.

Now that more terms have been computed, we can see that this is not the self-convolution of any integer sequence. - Paul D. Hanna, Aug 18 2006

Is it true that a(n+1)/a(n) tends to 4? Is there a heuristic argument why this might be true? - Gordon F. Royle, Aug 29 2006

Previous values a(30) = 5876236938019300 from Lorand Lucz, Jul 07 2013 and a(31) = 22974847474172100 from Lorand Lucz, Sep 03 2013 are wrong. New values a(30) and a(31) independently computed Kai Wang and Axel Kohnert. - Vaclav Kotesovec, Apr 15 2016

In the article by A. Iványi, G. Gombos, L. Lucz, T. Matuszka: "Parallel enumeration of degree sequences of simple graphs II" is in the tables on pages 258 and 261 a wrong value a(31) = 22974847474172100, but in the abstract another wrong value a(31) = 22974847474172374. - Vaclav Kotesovec, Apr 15 2016

The asymptotic formula given below confirms that a(n+1)/a(n) tends to 4. - Tom Johnston, Jan 18 2023

First differs from A000701 at a(11) = 28, A000701(11) = 27

A vertical section is a partial Young diagram with at most one square in each row.

Conjecture: a(n) is the number of non-half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

First differs from A046682 at a(11) = 28, A046682(11) = 29.

A vertical section is a partial Young diagram with at most one square in each row. For example, a suitable partition (shown as a coloring by positive integers) of the Young diagram of (322) is:

1 2 3

1 2

2 3

Conjecture: a(n) is the number of half-loop-graphical partitions of n. An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex, to be distinguished from a full loop, which has two equal vertices.

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as loop-graphical partitions (A339656). An integer partition is loop-graphical if it comprises the multiset of vertex-degrees of some graph with loops, where a loop is an edge with two equal vertices.

The following are equivalent characteristics for any positive integer n:

(1) the prime indices of n can be partitioned into distinct pairs, i.e. into a set of loops and edges;

(2) n can be factored into distinct semiprimes;

(3) the prime signature of n is loop-graphical.

In the covering case, these degree sequences, sorted in decreasing order, are the same thing as half-loop-graphical partitions (A321729). An integer partition is half-loop-graphical if it comprises the multiset of vertex-degrees of some graph with half-loops, where a half-loop is an edge with one vertex.

The following are equivalent characteristics for any positive integer n:

(1) the prime indices of n can be partitioned into distinct singletons or strict pairs, i.e., into a set of half-loops or edges;

(2) n can be factored into distinct primes or squarefree semiprimes;

(3) the prime signature of n is half-loop-graphical.