A027818 a(n) = (n+1)*binomial(n+6,6).

1, 14, 84, 336, 1050, 2772, 6468, 13728, 27027, 50050, 88088, 148512, 241332, 379848, 581400, 868224, 1268421, 1817046, 2557324, 3542000, 4834830, 6512220, 8665020, 11400480, 14844375, 19143306, 24467184, 31011904, 39002216, 48694800, 60381552, 74393088, 91102473

Offset: 0

Author

Thi Ngoc Dinh (via R. K. Guy)

Keywords

• nonn
• easy

Comments

Number of 13-subsequences of [ 1, n ] with just 6 contiguous pairs.

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
• Index entries for linear recurrences with constant coefficients, signature (8,-28,56,-70,56,-28,8,-1).

Crossrefs

Cf. A093564 ((7, 1) Pascal, column m=7). Partial sums of A050403.

Cf. A062190, A007318, A000142, A245334.

Programs

GAP

List([0..30], n-> (n+1)*Binomial(n+6,6)); # G. C. Greubel, Aug 29 2019

Haskell

a027818 n = (n + 1) * a007318' (n + 6) 6
-- Reinhard Zumkeller, Aug 31 2014

Magma

[(n+1)*Binomial(n+6,6): n in [0..30]]; // G. C. Greubel, Aug 29 2019

Maple

seq((n+1)*binomial(n+6,6), n=0..30); # Zerinvary Lajos, Oct 19 2006

Mathematica

Table[(n+1)*Binomial[n+6,6], {n,0,30}] (* G. C. Greubel, Aug 29 2019 *)

PARI

a(n) = (n+1)*binomial(n+6,6) \\ Charles R Greathouse IV, Jun 11 2015

Sage

[(n+1)*binomial(n+6,6) for n in (0..30)] # G. C. Greubel, Aug 29 2019

Formula

G.f.: (1+6*x)/(1-x)^8.

a(n) = A245334(n+6,6)/A000142(6). - Reinhard Zumkeller, Aug 31 2014

E.g.f.: (7! +9360*x +20520*x^2 +15000*x^3 +4650*x^4 +666*x^5 +43*x^6 + x^7)*exp(x)/7!. - G. C. Greubel, Aug 29 2019

From Amiram Eldar, Jan 28 2022: (Start)

Sum_{n>=0} 1/a(n) = Pi^2 - 5269/600.

Sum_{n>=0} (-1)^n/a(n) = Pi^2/2 - 512*log(2)/5 + 40189/600. (End)