A270229 Number of matchings in the 2 X n rook graph P_2 X K_n.

1, 2, 7, 32, 193, 1382, 11719, 112604, 1221889, 14639786, 192949639, 2760749048, 42732172993, 709490574158, 12596398359367, 237750425419508, 4757710386662401, 100516614496518866, 2236829315345704711, 52262526676903613264, 1279512810244450887361

Offset: 0

Author

Andrew Howroyd, Mar 13 2016

Keywords

• nonn

Comments

Sequence extended to n=0 using closed form. (binomial transform of A111883)

Links

• Andrew Howroyd, Table of n, a(n) for n = 0..50

Crossrefs

Cf. A270227, A270228, A000085, A081919 (perfect matchings).

Programs

Mathematica

a[n_] := Sum[Binomial[n, k]*Abs[HermiteH[k, I/Sqrt[2]]]^2/2^k, {k, 0, n}];
Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Oct 01 2017 *)
CoefficientList[Series[E^((2-x)*x/(1-x)) / Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Oct 01 2017 *)

Formula

Binomial transform of A111883.

From Vaclav Kotesovec, Oct 01 2017: (Start)

a(n) = (n+1)*a(n-1) + (n-1)^2*a(n-2) - (n-2)*(n-1)^2*a(n-3) + (n-3)*(n-2)*(n-1)*a(n-4).

E.g.f.: exp((2-x)*x/(1-x)) / sqrt(1-x^2).

a(n) ~ exp(1/2 + 2*sqrt(n) - n) * n^n / 2.

(End)