cp's OEIS Frontend

A370307 a(n) = A002623(n) + n.

%I A370307 #16 Feb 19 2024 17:11:49
%S A370307 1,4,9,16,26,39,56,77,103,134,171,214,264,321,386,459,541,632,733,844,
%T A370307 966,1099,1244,1401,1571,1754,1951,2162,2388,2629,2886,3159,3449,3756,
%U A370307 4081,4424,4786,5167,5568,5989,6431,6894,7379,7886,8416,8969,9546,10147,10773,11424,12101
%N A370307 a(n) = A002623(n) + n.
%C A370307 a(n) = number of left-labeled (2,n)-bipartite graphs.
%C A370307 The bipartite graphs counted here arise as representations of certain types of calls in switching networks in which two callers can be in a call with an arbitrary number (n) of receivers, and where callers are distinguishable, but the receivers are not. In a more abstract setting, a left-labeled (2,n)-bipartite graph is a graph with two sets of non-overlapping vertices I and O, where |I| = 2, |O| = n, and the two vertices in I are considered different (distinguishable), whereas the n vertices in O are considered interchangeable (indistinguishable). The sequence gives the number of non-isomorphic graphs under these assumptions.
%H A370307 A. Atmaca and A. Yavuz Oruc, <a href="https://arxiv.org/abs/2402.08053">On The Number Of Labeled Bipartite Graphs</a>, arXiv:2402.08053 [math.CO], 2024.
%H A370307 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (3,-2,-2,3,-1).
%F A370307 a(n) = (2*n^3 + 15*n^2 + 58*n + 45/2 + (3/2)*(-1)^n)/24.
%e A370307 For n = 2, suppose that the left vertices are distinguishable and labeled a and b. The right vertices are indistinguishable but labeled d and e for notational convenience to describe the edges in the example bipartite graphs.
%e A370307 The nine left-labeled (2,2)-bipartite graphs are
%e A370307 (1) Empty bipartite graph (no edges)
%e A370307 (2) Place an edge between a and d
%e A370307 (3) Place an edge between b and d.
%e A370307 (4) Place an edge between a and d and an edge between a and e.
%e A370307 (5) Place an edge between b and d and an edge between b and e
%e A370307 (6) Place an edge between a and d and an edge between b and d
%e A370307 (7) Place an edge between a and d and an edge between b and e
%e A370307 (8) Place an edge between a and d and an edge between b and (d and e)
%e A370307 (9) Place an edge between a and (d and e) and an edge between b and (d and e).
%e A370307 The provided formula works out as:  (2*2^3 + 15*2^2 + 58 * 2 + 22.5 + 1.5*(-1)^2)/24 = (16 + 60 + 116 + 24 )/24 = 216/24 = 9.
%t A370307 an = Function[n, (2 n^3 + 15 n^2 + 58 n + 45/2 + (3/2) (-1)^n)/(24)] /@ Range[0, 999, 1];
%t A370307 LinearRecurrence[{3, -2, -2, 3, -1}, {1, 4, 9, 16, 26}, 51] (* _Hugo Pfoertner_, Feb 15 2024 *)
%Y A370307 Cf. A002623.
%K A370307 nonn
%O A370307 0,2
%A A370307 _Yavuz Oruc_, Feb 14 2024
%E A370307 Edited by _N. J. A. Sloane_, Feb 19 2024 (simplified definition by referring to a classical sequence).