A027660 a(n) = C(n+2, 2) + C(n+2, 3) + C(n+2, 4) + C(n+2, 5).

1, 4, 11, 26, 56, 112, 210, 372, 627, 1012, 1573, 2366, 3458, 4928, 6868, 9384, 12597, 16644, 21679, 27874, 35420, 44528, 55430, 68380, 83655, 101556, 122409, 146566, 174406, 206336, 242792

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

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).

Crossrefs

Cf. A000295, A000332, A000389, A051601.

Programs

Magma

[(n^2-n+20)*Binomial(n+3,3)/20: n in [0..60]]; // G. C. Greubel, Aug 01 2022

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 *)

Sage

[binomial(n+3,5) +binomial(n+3,3) for n in range(0, 60)] # Zerinvary Lajos, May 17 2009

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

a(n) = A000389(n+5) - 2*A000332(n+3). - R. J. Mathar, 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)