A288035 Number of (undirected) paths in the complete bipartite graph K_n,n.

1, 12, 135, 2224, 55725, 2006316, 98309827, 6291829440, 509638185369, 50963818537900, 6166622043087231, 887993574204562992, 150070914040571147845, 29413899151951944980364, 6618127309189187620585275, 1694240591152432030869834496, 489635530843052856921382174257

Offset: 1

Author

Eric W. Weisstein, Jun 04 2017

Keywords

• nonn

Links

• Andrew Howroyd, Table of n, a(n) for n = 1..50
• Eric Weisstein's World of Mathematics, Complete Bipartite Graph
• Eric Weisstein's World of Mathematics, Graph Path

Crossrefs

Main diagonal of A307027 and A360850.

Cf. A070968, A048617, A010790.

Programs

Mathematica

Table[Sum[(n!)^2/((n - Ceiling[k/2])! (n - Floor[k/2])!), {k, 2, 2 n}], {n, 20}] (* Eric W. Weisstein, Jun 13 2017 *)
Table[n!^2 (BesselI[0, 2] + BesselI[1, 2] - HypergeometricPFQRegularized[{1}, {1 + n, 1 + n}, 1]) - n HypergeometricPFQ[{1}, {n, 1 + n}, 1], {n, 20}] // FunctionExpand (* Eric W. Weisstein, Jun 13 2017 *)

PARI

a(n) = sum(k=2, 2*n, n!^2/((n-(k+1)\2)!*(n-k\2)!)); \\ Andrew Howroyd, Jun 10 2017

PARI

a(n) = n!^2*sum(k=0, n-1, (1 + k)/(k!)^2) \\ Andrew Howroyd, Feb 24 2023

Formula

a(n) = Sum_{k=2..2*n} n!^2/((n-ceiling(k/2))!*(n-floor(k/2))!). - Andrew Howroyd, Jun 10 2017

a(n) = n!^2 * Sum_{k=0..n-1} (1 + k)/(k!^2). - Andrew Howroyd, Feb 24 2023

Extensions

Terms a(8) and beyond from Andrew Howroyd, Jun 10 2017