cp's OEIS Frontend

A283678 Number of possible draws of 2n pairs of consecutive cards from a set of 4n + 1 cards, so that the card that initially occupies the central position is not selected.

%I A283678 #36 Mar 29 2025 03:38:35
%S A283678 1,2,54,4500,771750,225042300,99843767100,62673358948200,
%T A283678 52880646612543750,57733914846094987500,79199384385873103852500,
%U A283678 133357363417740148141455000,270426506783940730406180497500,650063718230626755784087734375000,1827886309419060919156885553671875000
%N A283678 Number of possible draws of 2n pairs of consecutive cards from a set of 4n + 1 cards, so that the card that initially occupies the central position is not selected.
%C A283678 The probability that the middle card is not selected in a random draw of 2n consecutive card pairs between 4n + 1 cards is a(n)/(4n)!!.
%C A283678 Essentially a bisection of A001757. - _Giovanni Resta_, Mar 14 2017
%H A283678 Ignacio Larrosa Cañestro, <a href="https://groups.google.com/d/msg/es.ciencia.matematicas/Q2F1lCN9fI4/a30fnrpwBQAJ">El as de corazones</a>.
%F A283678 a(n) = binomial(2n, n)*((2n-1)!!)^2 = A092563(n)*A001147(n).
%F A283678 n*a(n) -2*(2*n-1)^3*a(n-1)=0. - _R. J. Mathar_, Jul 15 2017
%F A283678 a(n) ~ 2^(4*n+1)*(n/e)^(2*n)/sqrt(n*Pi). - _Stefano Spezia_, Mar 28 2025
%e A283678 For n = 1, you have 5 cards (A, B, C, D, E) and you can make 2 draws of pairs of consecutive cards (AB, DE) and (DE, AB) without select C.
%p A283678 ogf := sqrt(x) * BesselI(0, sqrt(x)/4) * BesselK(0,sqrt(x)/4) / 2;
%p A283678 simplify(subs(x=1/x, asympt(ogf, x, 20))); # _Mark van Hoeij_, Oct 24 2017
%t A283678 Table[Binomial[2n, n] Product[2n + 1 - 2i, {i, 1, n}]^2, {n, 0, 15}] (* _Indranil Ghosh_, Mar 22 2017 *)
%o A283678 (PARI) a(n)=binomial(2*n, n)*prod(i=1,n,2*n+1-2*i)^2 \\ _Charles R Greathouse IV_, Mar 14 2017
%o A283678 (Python)
%o A283678 from sympy import binomial, factorial2
%o A283678 print([binomial(2*n, n) * factorial2(2*n - 1)**2 for n in range(15)]) # _Indranil Ghosh_, Mar 22 2017
%Y A283678 Cf. A001147, A001757, A092563.
%K A283678 nonn,easy
%O A283678 0,2
%A A283678 _Ignacio Larrosa Cañestro_, Mar 14 2017