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 A338859 #40 Dec 23 2020 20:20:23
%S A338859 1,1,0,1,1,0,1,1,1,0,1,2,1,1,0,1,4,3,1,1,0,1,9,10,4,1,1,0,1,20,45,20,
%T A338859 6,1,1,0,1,48,210,165,55,8,1,1,0,1,115,1176,1540,1035,136,13,1,1,0,1,
%U A338859 286,6670,19600,22155,6273,430,18,1,1,0,1,719,41041,260130,692076,324008,46185,1300,30,1,1,0
%N A338859 Square array A(m,k) is the number of unicyclic graphs with m trees of k nodes; m,k >= 0, read by falling antidiagonals.
%C A338859 The number of unicyclic graphs with m k-trees is equal to the number of bracelets with m beads using up to A000081(k) colors, so A(m,k) = A321791(m, A000081(k)).
%C A338859 Because A102911(k) is the number of graphs constituted by 2 k-node rooted trees with the roots joined by an edge, A(2,k) = A102911(k). [Bomfim illustration for k=2,3].
%C A338859 Column 1 refers to Cyclic graphs, Column 2 refers to Sunlet graphs.
%H A338859 <a href="/index/Br#bracelets">Index entries for sequences related to bracelets</a>
%H A338859 Washington Bomfim, <a href="/A338859/a338859.png">Illustraction of graphs counted by A(2,k), k=2,3 </a>
%H A338859 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/SunletGraph.html">Sunlet graph</a>
%F A338859 A(m,k) = A321791(m, A000081(k)).
%e A338859 A begins,
%e A338859 ---+------------------------------------------------------------------------------
%e A338859 m/k|0 1 2 3 4 5 6 7 8 9
%e A338859 ---+------------------------------------------------------------------------------
%e A338859 0 |1 1 1 1 1 1 1 1 1 1 ...
%e A338859 1 |0 1 1 2 4 9 20 48 115 286 ...
%e A338859 2 |0 1 1 3 10 45 210 1176 6670 41041 ...
%e A338859 3 |0 1 1 4 20 165 1540 19600 260130 3939936 ...
%e A338859 4 |0 1 1 6 55 1035 22155 692076 22247785 842202361 ...
%e A338859 5 |0 1 1 8 136 6273 324008 25535712 2012117671 191362445560 ...
%e A338859 6 |0 1 1 13 430 46185 5376070 1020580232 192799298140 45606942211831 ...
%e A338859 7 |0 1 1 18 1300 344925 91508580 41936107248 19000229453710 11179807512382366 ...
%e A338859 ...| ... ... ... ... ...
%e A338859 ---+------------------------------------------------------------------------------
%e A338859 The A(3,3) = 4 unicyclic graphs with 3 trees of 3 nodes
%e A338859 0 0
%e A338859 | |
%e A338859 0 0 0 0 0 0
%e A338859 | \ / | \ /
%e A338859 0 0 0 0
%e A338859 /*\ /*\ /*\ /*\
%e A338859 /***\ /***\ /***\ /***\
%e A338859 0-----0 0---- 0 0-----0 0-----0
%e A338859 / \ / \ / \ / \ / \ | |
%e A338859 0 0 0 0 0 0 0 0 0 0 0 0
%e A338859 / \ | |
%e A338859 0 0 0 0
%e A338859 The graphs above are also representations of bracelets with m = 3 beads using up to A000081(k=3) = 2 colors.
%o A338859 (PARI) \\ From Robert A. Russell formula of A321791.
%o A338859 A(m, k)={ if( m == 0, return(1),
%o A338859 (k^((m+1)>>1)+k^ceil((m+1)/2)) / 4 + sumdiv(m, d, eulerphi(d)*k^(m/d) )/(m<<1)) };
%o A338859 seq(max_m) = { my(f = vector(max_m), kk, mm, ff); f[1] = 1;
%o A338859 for(j=1, max_m - 1, f[j+1] = 1/j * sum(k=1, j, sumdiv(k, d, d * f[d]) * f[j-k+1]));
%o A338859 print1(A(0,0) ", "); for(k = 1, max_m, kk = k; mm = 0; ff = f[kk];
%o A338859 until(A(mm,ff)==0, print1(A(mm,ff)", "); mm++; kk--; if(kk==0, ff=0, ff = f[kk]) );
%o A338859 print1("0, ")) };
%Y A338859 Cf. A000081 (row 1), A321791, A102911 (row 2), A000029 (column 3), A032275 (column 4).
%K A338859 nonn,tabl
%O A338859 0,12
%A A338859 _Washington Bomfim_, Nov 24 2020