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

0,2

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).

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)

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 *)

Prog

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

Crossrefs

Sequence in context: A013346 A013348 A013349 * A130610 A279582 A004339

Adjacent sequences: A140217 A140218 A140219 * A140221 A140222 A140223

Keyword

nonn,easy

Author

Zerinvary Lajos, Jun 23 2008

Status

approved