A027810 a(n) = (n+1)*binomial(n+5, 5).

1, 12, 63, 224, 630, 1512, 3234, 6336, 11583, 20020, 33033, 52416, 80444, 119952, 174420, 248064, 345933, 474012, 639331, 850080, 1115730, 1447160, 1856790, 2358720, 2968875, 3705156, 4587597, 5638528, 6882744, 8347680, 10063592, 12063744, 14384601, 17066028

Offset: 0

Author

Thi Ngoc Dinh (via R. K. Guy)

Keywords

• nonn
• easy

Comments

Number of 11-subsequences of [ 1, n ] with just 5 contiguous pairs.

References

• Albert H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.
• Herbert John Ryser, Combinatorial Mathematics, Carus Mathematical Monographs No. 14, John Wiley and Sons, 1963, pp. 1-8.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Crossrefs

Partial sums of A051843.

Cf. A093563 ((6, 1) Pascal, column m=6).

Cf. A007318, A000142, A245334.

Programs

Haskell

a027810 n = (n + 1) * a007318' (n + 5) 5
-- Reinhard Zumkeller, Aug 31 2014

Magma

[(n+1)*Binomial(n+5,5): n in [0..40]]; // Vincenzo Librandi, Jul 30 2014

Magma

[n*(n-1)*(n-2)*(n-3)*(n-4)^2/120: n in [5..40]]; // Vincenzo Librandi, Jul 30 2014

Maple

[seq(n*(n-1)*(n-2)*(n-3)*(n-4)^2/5!,n=5..33)]; # Zerinvary Lajos, Oct 19 2006

Mathematica

Table[(n+1)Binomial[n+5,5],{n,0,30}] (* Harvey P. Dale, Jul 29 2014 *)
CoefficientList[Series[(1 + 5 x)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Jul 30 2014 *)

PARI

a(n)=n*(n^5+16*n^4+100*n^3+310*n^2+499*n+394)/120+1 \\ Charles R Greathouse IV, Sep 28 2015

Formula

G.f.: (1+5*x)/(1-x)^7.

a(n) = A245334(n+5, 5)/A000142(5). - Reinhard Zumkeller, Aug 31 2014

From Amiram Eldar, Jan 28 2022: (Start)

Sum_{n>=0} 1/a(n) = 5*Pi^2/6 - 1025/144.

Sum_{n>=0} (-1)^n/a(n) = 5*Pi^2/12 - 160*log(2)/3 + 4865/144. (End)

Extensions

Two redundant formulas deleted by N. J. A. Sloane, Jul 30 2014