A140107 a(n) = binomial(n+3, 3)*7^n.

1, 28, 490, 6860, 84035, 941192, 9882516, 98825160, 951192165, 8877793540, 80787921214, 719746934452, 6297785676455, 54257845827920, 461191689537320, 3874010192113488, 32202709721943369, 265198785945415980, 2165790085220897170, 17554298585474640220

Offset: 0

Author

Zerinvary Lajos, Jun 03 2008

Keywords

• nonn
• easy

Comments

With a different offset, number of n-permutations (n=4) of 8 objects: s, t, u, v, w, z, x, y with repetition allowed, containing exactly three u's.

uuus, uusu, usuu, suuu,

uuut, uutu, utuu, tuuu,

uuuv, uuvu, uvuu, vuuu,

uuuw, uuwu, uwuu, wuuu,

uuuz, uuzu, uzuu, zuuu,

uuux, uuxu, uxuu, xuuu,

uuuy, uuyu, uyuu, yuuu

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (28,-294,1372,-2401).

Programs

Magma

[7^n* Binomial(n+3, 3): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Maple

seq(binomial(n+3,3)*7^n,n=0..26);

Mathematica

Table[Binomial[n+3,3]7^n,{n,0,20}] (* or *) LinearRecurrence[{28,-294,1372,-2401},{1,28,490,6860},20] (* Harvey P. Dale, Jun 21 2016 *)

PARI

a(n)=binomial(n+3,3)*7^n \\ Charles R Greathouse IV, Oct 07 2015

Formula

From R. J. Mathar, Jun 03 2009: (Start)

a(n) = 28*a(n-1) - 294*a(n-2) + 1372*a(n-3) - 2401*a(n-4).

G.f.: 1/(7*x-1)^4. (End)

From Amiram Eldar, Aug 28 2022: (Start)

Sum_{n>=0} 1/a(n) = 756*log(7/6) - 231/2.

Sum_{n>=0} (-1)^n/a(n) = 1344*log(8/7) - 357/2. (End)