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

%I A094219 #28 May 04 2025 01:57:41
%S A094219 0,0,0,0,7,112,1092,8400,56100,341088,1939938,10498488,54679625,
%T A094219 276276000,1362040680,6580248480,31256180280,146350008000,
%U A094219 676868787000,3097351569312,14042319855102,63144549413792,281895309883000
%N A094219 Number of permutations of length n with exactly 3 occurrences of the pattern 2-13.
%D A094219 R. Lie, Permutations and Patterns, Master's Thesis, Goeteborg, Sweden: Chalmers University of Technology, 2004.
%H A094219 Alois P. Heinz, <a href="/A094219/b094219.txt">Table of n, a(n) for n = 1..500</a>
%H A094219 Robert Parviainen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL9/Parviainen/parviainen3.html">Lattice Path Enumeration of Permutations with k Occurrences of the Pattern 2-13</a>, Journal of Integer Sequences, Vol. 9 (2006), Article 06.3.2.
%F A094219 a(n) = (1/3)*binomial(n+2,2)*binomial(2*n,n-5).
%F A094219 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
%F A094219 From _Amiram Eldar_, May 04 2025: (Start)
%F A094219 Sum_{n>=5} 1/a(n) = sqrt(3)*Pi/2 - 719/280.
%F A094219 Sum_{n>=5} (-1)^(n+1)/a(n) = 129*sqrt(5)*log(phi) - 194141/1400, where phi is the golden ratio (A001622). (End)
%t A094219 Table[Binomial[n + 2, 2] Binomial[2 n, n - 5]/3, {n, 1, 30}] (* _Vincenzo Librandi_, Aug 20 2015 *)
%o A094219 (PARI) a(n)=1/3*binomial(n+2,2)*binomial(2*n,n-5)
%o A094219 (Magma) [(1/3)*Binomial(n+2,2)*Binomial(2*n,n-5): n in [1..30]]; // _Vincenzo Librandi_, Aug 20 2015
%Y A094219 Cf. A094218, A001622.
%Y A094219 Column k=3 of A263776.
%K A094219 nonn
%O A094219 1,5
%A A094219 _Benoit Cloitre_, May 27 2004