A140220 a(n) = binomial(n+7, 7)*5^n.

1, 40, 900, 15000, 206250, 2475000, 26812500, 268125000, 2513671875, 22343750000, 189921875000, 1553906250000, 12301757812500, 94628906250000, 709716796875000, 5204589843750000, 37407989501953125, 264056396484375000, 1833724975585937500, 12546539306640625000

Offset: 0

Author

Zerinvary Lajos, Jun 23 2008

Keywords

• nonn
• easy

Comments

With a different offset, number of n-permutations (n>=7) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly seven (7) u's.

If n=7 then a(0)=1.

Example: a(1)=40 because we have

uuuuuuut, uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuuy,

uuuuuutu, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuuyu,

uuuuutuu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuuyuu,

uuuutuuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuuyuuu,

uuutuuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuuyuuuu,

uutuuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uuyuuuuu,

utuuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, uyuuuuuu,

tuuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu, yuuuuuuu.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000
• Index entries for linear recurrences with constant coefficients, signature (40,-700,7000,-43750,175000,-437500,625000,-390625).

Programs

Magma

[Binomial(n+7, 7)*5^n: n in [0..20]]; // Vincenzo Librandi, Feb 09 2018

Maple

seq(binomial(n+7,7)*5^n,n=0..18);

Mathematica

Table[Binomial[n+7,7]*5^n,{n,0,20}] (* Harvey P. Dale, Oct 07 2014 *)
CoefficientList[Series[1 / (1 - 5 x)^8, {x, 0, 33}], x] (* Vincenzo Librandi, Feb 09 2018 *)

Formula

From Chai Wah Wu, Mar 20 2017: (Start)

a(n) = 40*a(n-1) - 700*a(n-2) + 7000*a(n-3) - 43750*a(n-4) + 175000*a(n-5) - 437500*a(n-6) + 625000*a(n-7) - 390625*a(n-8) for n > 7.

G.f.: 1/(1 - 5*x)^8. (End)

From Amiram Eldar, Aug 29 2022: (Start)

Sum_{n>=0} 1/a(n) = 143360*log(5/4) - 191933/6.

Sum_{n>=0} (-1)^n/a(n) = 1632960*log(6/5) - 1786337/6. (End)