A297470 Number of maximal matchings in the n-barbell graph.

1, 2, 9, 18, 225, 450, 11025, 22050, 893025, 1786050, 108056025, 216112050, 18261468225, 36522936450, 4108830350625, 8217660701250, 1187451971330625, 2374903942661250, 428670161650355625, 857340323300711250, 189043541287806830625, 378087082575613661250

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..100

Eric Weisstein's World of Mathematics, Barbell Graph

Eric Weisstein's World of Mathematics, Matching

Eric Weisstein's World of Mathematics, Maximal Independent Edge Set

Formula

a(2n) = 2*(2*n-1)!!^2, a(2n-1) = (2*n-1)!!^2. - Andrew Howroyd, Dec 30 2017

D-finite with recurrence: (n-3)*(n+1)*a(n) + (4*n-2)*a(n-1) - (n+2)*(n-2)*(n-1)^2*a(n-2) = 0. - Georg Fischer, Nov 25 2022

Mathematica

Table[2^(n + 1) ((n - 1 + Mod[n, 2])/2)!^2/Pi, {n, 20}]
Table[2^(n + 1) ((2 Ceiling[n/2] - 1)/2)!^2/Pi, {n, 20}]
Table[If[Mod[n, 2] == 0, 2 (n - 1)!!^2, n!!^2], {n, 20}]
a[ n_] := If[n<0, 0, With[{t = Ceiling[n/2]}, (n+2-2*t)*((2*t)!/(2^t*t!))^2]]; (* Michael Somos, May 27 2023 *)

Prog

(PARI) \\ here b(n)=A001147(n)=(2*n-1)!!
b(n)={(2*n)!/(2^n*n!)}
a(n)=if(n%2, b((n+1)/2)^2, 2*b(n/2)^2); \\ Andrew Howroyd, Dec 30 2017
(PARI) {a(n) = if(n<1, 0, my(t = (n+1)\2); (n+2-2*t)*((2*t)!/(2^t*t!))^2)}; /* Michael Somos, May 27 2023 */

Crossrefs

Cf. A001147, A288841, A290715, A297007.

Sequence in context: A068978 A006226 A109298 * A075537 A342473 A342619

Adjacent sequences: A297467 A297468 A297469 * A297471 A297472 A297473

Keyword

nonn

Author

Eric W. Weisstein, Dec 30 2017

Extensions

a(1)-a(2) and terms a(21) and beyond from Andrew Howroyd, Dec 30 2017

Status

approved