This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A138388 #13 Sep 10 2024 16:14:01
%S A138388 1,2,8,27,89,289,938,2985,9456,29722,92842,288509,892506,2749940,
%T A138388 8443504,25845735,78897469,240259510,730040882,2213910727,6701939407,
%U A138388 20255424836,61128717826,184233427305,554574518396,1667491889239
%N A138388 Numbers of unlabeled graphs with n vertices and 3 unicyclic components.
%C A138388 This sequence is the third row of table T of A137918.
%C A138388 Let us refer to a partition of n that has exactly x parts as an x-partition of n.
%C A138388 A138387(n-3) counts the graphs corresponding to the 3-partitions of n whose smallest part is 3, so we consider only the 3-partitions of n whose smallest part is 4.
%C A138388 To determine those partitions, we start with k = 4, 5, ..., floor(n/3), and append to each one of these values of k the 2-partitions of n-k whose smallest part is >= k.
%C A138388 For example, if n = 18, we have 4 <= k <= 6. For k = 4, the 3-partitions are 4+(4+10), 4+(5+9), 4+(6+8) and 4+(7+7). To k = 5 correspond 5+(5+8) and 5+(6+7). Finally we have 6+(6+6).
%C A138388 To determine the formula, one must consider that there are partitions having distinct parts, partitions having 2 equal parts and if n mod 3 = 0, there is a unique partition with equal parts. See example.
%H A138388 D. E. Knuth, <a href="http://www-cs-faculty.stanford.edu/~knuth/fasc4b.ps.gz">The Art of Computer Programming</a>, Vol. 4, Fascicle 4b, p. 20.
%F A138388 If n <= 11, a(n) = A138387(n-3).
%F A138388 If n > 11, a(n) = A138387(n-3) + nU + sI + sD + sF, where nU = (f(n/3) + 2) * (f(n/3) + 1) * f(n/3)/6, (n mod 3 = 0), 0, (otherwise),
%F A138388 sI = (1/2) * Sum_{4 <= k < n/3}(f(k) + 1) * f(k) * f(n - 2k),
%F A138388 sD = Sum_{4 <= k < (n-2)/3} f(k) * Sum_{k+1 <= i < (n-k)/2} f(i) * f(n-k-i),
%F A138388 sF = (1/2) * Sum_{4 <= k < n/3}f(k) * (f((n-k)/2) + 1) * f((n-k)/2), (even n, k), or (1/2) * Sum_{5 <= k < n/3}f(k) * (f((n-k)/2) + 1) * f((n-k)/2), (odd n, k),
%F A138388 where f(j) is A001429(j).
%e A138388 a(18) = 29722, since A138387(15) = 17980; nU = 455. The partitions considered by sI are 4+(4+10) and 5+(5+8). Those partitions correspond to 3306 graphs. To sD correspond 4+(5+9), 4+(6+8) and 5+(6+7). This gives 6859 graphs. To sF correspond 4+(7+7), or 1122 graphs.
%e A138388 Note that f(4)= 2, f(5) = 5, f(6) = 13, f(7) = 33, f(8) = 89, f(9) = 240 and f(10) = 657.
%Y A138388 Cf. A001429, A137918.
%K A138388 nonn
%O A138388 9,2
%A A138388 _Washington Bomfim_, Mar 18 2008