|
%I #37 Jun 12 2019 11:20:29
%S 1,6,25,88,280,832,2352,6400,16896,43520,109824,272384,665600,1605632,
%T 3829760,9043968,21168128,49152000,113311744,259522560,590872576,
%U 1337982976,3014656000,6761218048,15099494400,33587986432,74440507392
%N Second column of triangle A055584.
%C Number of 132-avoiding permutations of [n+5] containing exactly three 123 patterns. - _Emeric Deutsch_, Jul 13 2001
%C If X_1,X_2,...,X_n are 2-blocks of a (2n+2)-set X then, for n>=1, a(n-1) is the number of (n+3)-subsets of X intersecting each X_i, (i=1,2,...,n). - _Milan Janjic_, Nov 18 2007
%C Convolution of A001792 with itself. - _Philippe Deléham_, Feb 21 2013
%H Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Two Enumerative Functions</a>
%H M. Janjic, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL11/Janjic/janjic19.html">On a class of polynomials with integer coefficients</a>, JIS 11 (2008) 08.5.2
%H Pudwell, Lara; Scholten, Connor; Schrock, Tyler; Serrato, Alexa <a href="https://doi.org/10.1155/2014/316535">Noncontiguous pattern containment in binary trees</a>, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Section 5.2.
%H A. Robertson, H. S. Wilf and D. Zeilberger, <a href="http://www.combinatorics.org/ojs/index.php/eljc/article/view/v6i1r38">Permutation patterns and continued fractions,</a> Electr. J. Combin. 6, 1999, #R38.
%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (8,-24,32,-16).
%F G.f.: (1-x)^2/(1-2*x)^4.
%F a(n) = A055584(n+1, 1). a(n) = sum(a(j), j=0..n-1)+A001793(n+1), n >= 1.
%F a(n) = 2^(n-3)(n+1)(n+3)(n+8)/3.
%F Preceded by 0, this is the binomial transform of the tetrahedral numbers A000292. - _Carl Najafi_, Sep 08 2011
%F E.g.f.: (1/6)*(2*x^3 + 15*x^2 + 24*x + 6)*exp(2*x). - _G. C. Greubel_, Aug 22 2015
%e a(1)=6 because 432516,432561,435126,452136,532146 and 632145 are the only 132-avoiding permutations of 123456, containing exactly three increasing subsequences of length 3.
%t Table[(1/3)*2^(n-3)*(n+1)*(n+3)*(n+8), {n,0,50}] (* _G. C. Greubel_, Aug 22 2015 *)
%t LinearRecurrence[{8,-24,32,-16},{1,6,25,88},30] (* _Harvey P. Dale_, Nov 03 2017 *)
%o (PARI) Vec(((1-x)^2)/(1-2*x)^4 + O(x^30)) \\ _Michel Marcus_, Aug 22 2015
%Y Cf. A055584, partial sums of A049612, n >= 1.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_, May 26 2000
|
