A094219 Number of permutations of length n with exactly 3 occurrences of the pattern 2-13.

0, 0, 0, 0, 7, 112, 1092, 8400, 56100, 341088, 1939938, 10498488, 54679625, 276276000, 1362040680, 6580248480, 31256180280, 146350008000, 676868787000, 3097351569312, 14042319855102, 63144549413792, 281895309883000

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. A094218, A001622.

Column k=3 of A263776.

Programs

Magma

[(1/3)*Binomial(n+2,2)*Binomial(2*n,n-5): n in [1..30]]; // Vincenzo Librandi, Aug 20 2015

Mathematica

Table[Binomial[n + 2, 2] Binomial[2 n, n - 5]/3, {n, 1, 30}] (* Vincenzo Librandi, Aug 20 2015 *)

PARI

a(n)=1/3*binomial(n+2,2)*binomial(2*n,n-5)

Formula

a(n) = (1/3)*binomial(n+2,2)*binomial(2*n,n-5).

G.f.: (-7*x^2+8*x-2-(4*x^7+14*x^6+84*x^5-350*x^4+350*x^3-147*x^2+28*x-2) /(1-4*x)^(5/2)) /(2*x^5). - Mark van Hoeij, Apr 30 2013

From Amiram Eldar, May 04 2025: (Start)

Sum_{n>=5} 1/a(n) = sqrt(3)*Pi/2 - 719/280.

Sum_{n>=5} (-1)^(n+1)/a(n) = 129*sqrt(5)*log(phi) - 194141/1400, where phi is the golden ratio (A001622). (End)