A270229 - OEIS

%I #22 Aug 04 2024 15:13:17

%S 1,2,7,32,193,1382,11719,112604,1221889,14639786,192949639,2760749048,

%T 42732172993,709490574158,12596398359367,237750425419508,

%U 4757710386662401,100516614496518866,2236829315345704711,52262526676903613264,1279512810244450887361

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

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

%H Andrew Howroyd, <a href="/A270229/b270229.txt">Table of n, a(n) for n = 0..50</a>

%F Binomial transform of A111883.

%F From _Vaclav Kotesovec_, Oct 01 2017: (Start)

%F 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).

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

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

%F (End)

%t a[n_] := Sum[Binomial[n, k]*Abs[HermiteH[k, I/Sqrt[2]]]^2/2^k, {k, 0, n}];

%t Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Oct 01 2017 *)

%t CoefficientList[Series[E^((2-x)*x/(1-x)) / Sqrt[1-x^2], {x, 0, 20}], x] * Range[0, 20]! (* _Vaclav Kotesovec_, Oct 01 2017 *)

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

%K nonn

%O 0,2

%A _Andrew Howroyd_, Mar 13 2016