A115047 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.

1, 1, 5, 61, 1379, 49946, 2648967, 193530835, 18634276859, 2286742481794, 348390662991293, 64519134394428000, 14273926322439378685, 3718118808742139574436, 1126348335942168962657751, 392634641364638381277506199, 156052858498185218872911914627, 70147998632789834910508237254650

Offset

3,3

Comments

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

The sequence is denoted by V_n in Bertoldi, et al. 2004 in equation (2.27) on page 13. - Michael Somos, Sep 20 2014

I conjecture that a(n) is divisible by n if n is a prime. - Michael Somos, Sep 20 2014

Links

Seiichi Manyama, Table of n, a(n) for n = 3..246

Per Alexandersson and Linus Jordan, Enumeration of border-strip decompositions, Journal of Integer Sequences, Vol. 22 (2019), Article 19.4.5.

G. Bertoldi, S. Bolognesi, M. Matone, L. Mazzucato, and Y. Nakayama, The Liouville Geometry of N=2 Instantons and the Moduli of Punctured Spheres, arXiv:hep-th/0405117, 2004.

Yinsong Chen and Vladislav Kargin, On enumeration and entropy of ribbon tilings, The Electronic Journal of Combinatorics 30(2) (2023), #P2.15. See p. 3.

R. Kaufmann, Yu. Manin, and D. Zagier, Higher Weil-Petersson Volumes of Moduli Spaces of Stable n-pointed Curves, arXiv:alg-geom/9604001, 1996.

Formula

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

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

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

Example

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);
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!) +- ....

Maple

A115047 := proc(n)
   option remember;
   if n = 3 then
       1;
   else
       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)
     end if;
end proc:
seq(A115047(n), n=3..20) ; # R. J. Mathar, Nov 12 2011

Mathematica

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 *)
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 *)
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 *)

Prog

(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
(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))}
for(n=3, 25, print1(a(n-2), ", ")) \\ Paul D. Hanna, Oct 01 2014

Crossrefs

Sequence in context: A345103 A367385 A096537 * A000364 A028296 A159316

Adjacent sequences: A115044 A115045 A115046 * A115048 A115049 A115050

Keyword

nonn,easy

Author

N. J. A. Sloane, based on an email message from John McKay, Feb 28 2006

Status

approved