A094218 Number of permutations of length n with exactly 2 occurrences of the pattern 2-13.

0, 0, 0, 2, 25, 198, 1274, 7280, 38556, 193800, 937992, 4412826, 20309575, 91861770, 409704750, 1806342720, 7887861960, 34166674800, 146977222320, 628521016500, 2673950235138, 11324837666604, 47773836727540, 200828153398752

Offset: 1

Author

Benoit Cloitre, May 27 2004

Keywords

• nonn

References

• R. Lie, Permutations and Patterns, Master's Thesis, Goeteborg, Sweden: Chalmers University of Technology, 2004.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..500
• Robert Parviainen, Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.

Crossrefs

Cf. A094219, A001622.

Column k=2 of A263776.

Programs

Magma

[n*Binomial(2*n,n-4)/2: n in [1..30]]; // Vincenzo Librandi, Aug 20 2015

Mathematica

Table[n Binomial[2 n, n - 4]/2, {n, 30}] (* Vincenzo Librandi, Aug 20 2015 *)

PARI

a(n)=n*binomial(2*n,n-4)/2

Formula

a(n) = n*binomial(2*n,n-4)/2.

From Amiram Eldar, May 04 2025: (Start)

Sum_{n>=4} 1/a(n) = 1147/45 - 34*Pi/(3*sqrt(3)) - 4*Pi^2/9.

Sum_{n>=4} (-1)^n/a(n) = 16*log(phi)^2 + 1588*log(phi)/(3*sqrt(5)) - 5272/45, where phi is the golden ratio (A001622). (End)