A217523 Number of assembly trees for complete bipartite graph K_{n,n}.

0, 1, 10, 450, 46440, 8580600, 2485501200, 1038647610000, 591422976144000, 440175696904944000, 414834426527320800000, 482828797838174467680000, 680160665982184667280000000, 1140497273795065245115056000000, 2244756232031112064775686176000000

Offset

0,3

Links

Table of n, a(n) for n=0..14.

Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, arXiv preprint, arXiv:1204.3842 [math.CO], 2012. See Theorem 23.

Andrew Vince and Miklos Bona, The Number of Ways to Assemble a Graph, The Electronic Journal of Combinatorics, Volume 19, Issue 4 (2012), Article P54.

Eric Weisstein's World of Mathematics, Assembly Number

Eric Weisstein's World of Mathematics, Complete Bipartite Graph

Formula

Define b(n) = (2*(6*n^2 - 12*n + 5)/n^2)*b(n-1) - ((4*n^3 - 20*n^2 + 29*n - 10)/(n^3 - n^2))*b(n-2), with b(0) = 0, b(1) = 1 and b(2) = 5/2. Then a(n) = n!*n!*b(n). - Franck Maminirina Ramaharo, Jan 28 2019

Mathematica

Table[n!^2 SeriesCoefficient[1 - Sqrt[(1 - x)^2 + (1 - y)^2 - 1], {x, 0, n}, {y, 0, n}], {n, 10}] (* Eric W. Weisstein, Mar 01 2023 *)
With[{nterms = 10}, RecurrenceTable[{b[1] == 1, b[2] == 5/2, b[n] == 2 (6 n^2 - 12 n + 5)/n^2 b[n - 1] - (4 n^3 - 20 n^2 + 29 n - 10)/(n^3 - n^2) b[n - 2]}, b, {n, nterms}] Range[nterms]!^2] (* Eric W. Weisstein, Mar 01 2023 *)

Prog

(Maxima)
(b[0] : 0, b[1] : 1, b[2] : 5/2, b[n] := (2*(6*n^2 - 12*n + 5)/n^2)*b[n - 1] - ((4*n^3 - 20*n^2 + 29*n - 10)/(n^3 - n^2))*b[n - 2])$
a(n) := n!*n!*b[n]$ /* Franck Maminirina Ramaharo, Jan 28 2019 */

Crossrefs

Cf. A361072 (number of assembly trees for K_{n,n,n}).

Sequence in context: A222665 A177391 A304289 * A232773 A221043 A337757

Adjacent sequences: A217520 A217521 A217522 * A217524 A217525 A217526

Keyword

nonn

Author

N. J. A. Sloane, Oct 08 2012

Extensions

More terms from Franck Maminirina Ramaharo, Jan 28 2019

Status

approved