OFFSET
0,2
COMMENTS
Number of 132-avoiding permutations of [n+5] containing exactly three 123 patterns. - Emeric Deutsch, Jul 13 2001
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
Convolution of A001792 with itself. - Philippe Deléham, Feb 21 2013
LINKS
Amiram Eldar, Table of n, a(n) for n = 0..1000
Milan Janjić, Two Enumerative Functions.
Milan Janjić, On a class of polynomials with integer coefficients, JIS 11 (2008), Article 08.5.2.
Lara Pudwell, Connor Scholten, Tyler Schrock, and Alexa Serrato, Noncontiguous pattern containment in binary trees, ISRN Comb. 2014, Article ID 316535, 8 p. (2014), Section 5.2.
Aaron Robertson, Herbert S. Wilf, and Doron Zeilberger, Permutation patterns and continued fractions, Electr. J. Combin. 6 (1999), Article R38.
Index entries for linear recurrences with constant coefficients, signature (8,-24,32,-16).
FORMULA
G.f.: (1-x)^2/(1-2*x)^4.
a(n) = A055584(n+1, 1).
a(n) = Sum_{j=0..n-1} a(j) + A001793(n+1), n >= 1.
a(n) = 2^(n-3)*(n+1)*(n+3)*(n+8)/3.
Preceded by 0, this is the binomial transform of the tetrahedral numbers A000292. - Carl Najafi, Sep 08 2011
E.g.f.: (1/6)*(2*x^3 + 15*x^2 + 24*x + 6)*exp(2*x). - G. C. Greubel, Aug 22 2015
From Amiram Eldar, Nov 16 2025: (Start)
Sum_{n>=0} 1/a(n) = 5592*log(2)/35 - 134164/1225.
Sum_{n>=0} (-1)^n/a(n) = 96084/1225 - 6696*log(3/2)/35. (End)
EXAMPLE
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.
MATHEMATICA
Table[(1/3)*2^(n-3)*(n+1)*(n+3)*(n+8), {n, 0, 50}] (* G. C. Greubel, Aug 22 2015 *)
LinearRecurrence[{8, -24, 32, -16}, {1, 6, 25, 88}, 30] (* Harvey P. Dale, Nov 03 2017 *)
PROG
(PARI) Vec(((1-x)^2)/(1-2*x)^4 + O(x^30)) \\ Michel Marcus, Aug 22 2015
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Wolfdieter Lang, May 26 2000
EXTENSIONS
More terms from Amiram Eldar, Nov 16 2025
STATUS
approved