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%I #65 Apr 28 2023 12:18:32
%S 1,1,5,61,1379,49946,2648967,193530835,18634276859,2286742481794,
%T 348390662991293,64519134394428000,14273926322439378685,
%U 3718118808742139574436,1126348335942168962657751,392634641364638381277506199,156052858498185218872911914627,70147998632789834910508237254650
%N a(3)=1; a(n) = Sum_{i=1..n-3} binomial(n-4,i-1)*binomial(n,i+1)*a(i+2)*a(n-i)*i*(n-i-2)/(2*(n-1)) for n > 3.
%C a(n) is the number of ways to tile a 2n X n rectangle with border strips of size n. This set is in bijection with permutations of x1,...,xn, y1,...,yn such that xi always appears before yi and if i>j then xi is not adjacent to yj. - _Per Alexandersson_, May 26 2018
%C The sequence is denoted by V_n in Bertoldi, et al. 2004 in equation (2.27) on page 13. - _Michael Somos_, Sep 20 2014
%C I conjecture that a(n) is divisible by n if n is a prime. - _Michael Somos_, Sep 20 2014
%H Seiichi Manyama, <a href="/A115047/b115047.txt">Table of n, a(n) for n = 3..246</a>
%H Per Alexandersson and Linus Jordan, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL22/Alexandersson/alex4.html">Enumeration of border-strip decompositions</a>, Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.5.
%H G. Bertoldi, S. Bolognesi, M. Matone, L. Mazzucato, and Y. Nakayama, <a href="https://arxiv.org/abs/hep-th/0405117">The Liouville Geometry of N=2 Instantons and the Moduli of Punctured Spheres</a>, arXiv:hep-th/0405117, 2004.
%H Yinsong Chen and Vladislav Kargin, <a href="https://doi.org/10.37236/10991">On enumeration and entropy of ribbon tilings</a>, The Electronic Journal of Combinatorics 30(2) (2023), #P2.15. See p. 3.
%H R. Kaufmann, Yu. Manin, and D. Zagier, <a href="https://arxiv.org/abs/alg-geom/9604001">Higher Weil-Petersson Volumes of Moduli Spaces of Stable n-pointed Curves</a>, arXiv:alg-geom/9604001, 1996.
%F If g(x) = Sum_{k>1} a(k+1) / (k * (k-2)!^2) * x^k, then 0 = x*(x - g(x))*g''(x) + x*g'(x)^2 - (x - g(x))*g'(x). [Bertoldi, et al. equation (2.31) page 14] - _Michael Somos_, Sep 20 2014
%F If y(x) = Sum_{k>0} a(k+2) / (k! * (k-1)!) * x^k, then x(y) = Sum_{k>0} -(-1)^k / (k! * (k-1)!) y^k. [Bertoldi, et al. equation (2.37) page 14] - _Michael Somos_, Sep 20 2014
%F Asymptotics (Kaufmann, et al., 1996, page 3): a(n) ~ b * (2*n-6)! / C^(n-3), where C = 2.49691833939101330106869449429726103117269753436051258..., b = 1.362053722455447392992552565765491313... . - _Vaclav Kotesovec_, Oct 04 2014
%e E.g.f.: A(x) = x + x^2/(1!*2!) + 5*x^3/(2!*3!) + 61*x^4/(3!*4!) + 1379*x^5/(4!*5!) + 49946*x^6/(5!*6!) + ... (if offset 1);
%e where Series_Reversion(A(x)) = x - x^2/(1!*2!) + x^3/(2!*3!) - x^4/(3!*4!) + x^5/(4!*5!) - x^6/(5!*6!) +- ....
%p A115047 := proc(n)
%p option remember;
%p if n = 3 then
%p 1;
%p else
%p add( binomial(n-4,i-1) *binomial(n,i+1) *procname(i+2) *procname(n-i) *i *(n-i-2) /(n-1)/2, i =1..n-3)
%p end if;
%p end proc:
%p seq(A115047(n),n=3..20) ; # _R. J. Mathar_, Nov 12 2011
%t a[3] = 1; a[n_] := a[n] = Sum[Binomial[n-4, i-1]*Binomial[n, i+1]*a[i+2]*a[n-i]*i*(n-i-2)/(2*(n-1)), {i, 1, n-3}]; Table[a[n], {n, 3, 20}] (* _Jean-François Alcover_, Mar 20 2014 *)
%t a[ n_] := With[{m = n - 2}, If[ m < 1, 0, m! (m - 1)! SeriesCoefficient[ InverseSeries[ Series[Integrate[ BesselJ[ 0, 2 Sqrt[x]], x], {x, 0, m}]], {x, 0, m}]]]; (* _Michael Somos_, Sep 20 2014 *)
%t a[n_] := Sum[ (-1)^Tr[p - 1] 1/(Times @@ ((Last /@ Tally[p])!)) (Multinomial @@ p) (Multinomial @@ (p + 1)), {p, IntegerPartitions[n-3]}]; (* _Per W. Alexandersson_, Jun 24 2020 *)
%o (PARI) v=vector(30);v[3]=1;for(n=4,#v,v[n]=sum(i=1,n-3,binomial(n-4,i-1)*binomial(n,i+1)*v[i+2]*v[n-i]*i*(n-i-2)/(n-1)/2));v \\ _Charles R Greathouse IV_, Nov 08 2011
%o (PARI) {a(n)=if(n<1,0,n!*(n-1)!*polcoeff(serreverse(sum(m=1,n,-(-x)^m/m!/(m-1)! +x*O(x^n))),n))}
%o for(n=3,25,print1(a(n-2),", ")) \\ _Paul D. Hanna_, Oct 01 2014
%K nonn,easy
%O 3,3
%A _N. J. A. Sloane_, based on an email message from John McKay, Feb 28 2006
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