OFFSET
0,2
COMMENTS
Also, number of 135246-avoiding permutations of n+2 with exactly 1 descent. E.g., there are 57 permutations of 6 with exactly 1 descent. Of these, only the permutation 135246 contains the pattern 135246 so a(4)=56. - Mike Zabrocki, Nov 29 2004
If Y is a 2-subset of an n-set X then, for n>=5, a(n-5) is the number of 5-subsets of X which do not have exactly one element in common with Y. - Milan Janjic, Dec 28 2007
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..1000
Alexsandar Petojevic, The Function vM_m(s; a; z) and Some Well-Known Sequences, Journal of Integer Sequences, Vol. 5 (2002), Article 02.1.7.
Index entries for linear recurrences with constant coefficients, signature (6,-15,20,-15,6,-1).
FORMULA
a(n) = (n+3)*(n+2)*(n+1)*(n^2 - n + 20)/120.
G.f.: (1 - 2*x + 2*x^2)/(1-x)^6. - Mike Zabrocki, Nov 29 2004
a(n) = binomial(n+3,5) + binomial(n+3,3). - Zerinvary Lajos, Jul 24 2006, corrected Oct 01 2021
From G. C. Greubel, Aug 01 2022: (Start)
a(n) = Sum_{j=0..3} binomial(n+2, j+2).
E.g.f.: (1/120)*(120 +360*x +240*x^2 +80*x^3 +15*x^4 +x^5)*exp(x). (End)
MAPLE
a:= n-> (n+3)*(n+2)*(n+1)*(n^2-n+20)/120;
seq(a(n), n = 0..60);
MATHEMATICA
Sum[Binomial[3+Range[0, 60], 2*j+1], {j, 2}] (* G. C. Greubel, Aug 01 2022 *)
PROG
(Sage) [binomial(n+3, 5) +binomial(n+3, 3) for n in range(0, 60)] # Zerinvary Lajos, May 17 2009
(Magma) [(n^2-n+20)*Binomial(n+3, 3)/20: n in [0..60]]; // G. C. Greubel, Aug 01 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
STATUS
approved