A140346 a(n) = binomial(n+8, 8)*5^n.

1, 45, 1125, 20625, 309375, 4021875, 46921875, 502734375, 5027343750, 47480468750, 427324218750, 3690527343750, 30754394531250, 248400878906250, 1951721191406250, 14963195800781250, 112223968505859375, 825176239013671875, 5959606170654296875, 42344570159912109375

Offset

0,2

Comments

With a different offset, number of n-permutations (n>=8) of 6 objects: t, u, v, z, x, y with repetition allowed, containing exactly eight (8) u's.

If n=8 then a(0)=1.

Example: a(1)=45 because we have

uuuuuuuut, uuuuuuuuv, uuuuuuuuz, uuuuuuuux, uuuuuuuuy,

uuuuuuutu, uuuuuuuvu, uuuuuuuzu, uuuuuuuxu, uuuuuuuyu,

uuuuuutuu, uuuuuuvuu, uuuuuuzuu, uuuuuuxuu, uuuuuuyuu,

uuuuutuuu, uuuuuvuuu, uuuuuzuuu, uuuuuxuuu, uuuuuyuuu,

uuuutuuuu, uuuuvuuuu, uuuuzuuuu, uuuuxuuuu, uuuuyuuuu,

uuutuuuuu, uuuvuuuuu, uuuzuuuuu, uuuxuuuuu, uuuyuuuuu,

uutuuuuuu, uuvuuuuuu, uuzuuuuuu, uuxuuuuuu, uuyuuuuuu,

utuuuuuuu, uvuuuuuuu, uzuuuuuuu, uxuuuuuuu, uyuuuuuuu,

tuuuuuuuu, vuuuuuuuu, zuuuuuuuu, xuuuuuuuu, yuuuuuuuu.

Links

Table of n, a(n) for n=0..19.

Index entries for linear recurrences with constant coefficients, signature (45,-900,10500,-78750,393750,-1312500,2812500,-3515625,1953125).

Formula

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

a(n) = 45*a(n-1) - 900*a(n-2) + 10500*a(n-3) - 78750*a(n-4) + 393750*a(n-5) - 1312500*a(n-6) + 2812500*a(n-7) - 3515625*a(n-8) + 1953125*a(n-9) for n > 8.

G.f.: 1/(1 - 5*x)^9. (End)

From Amiram Eldar, Aug 29 2022: (Start)

Sum_{n>=0} 1/a(n) = 11197440*log(6/5) - 14290736/7.

Sum_{n>=0} (-1)^n/a(n) = 3071048/21 - 655360*log(5/4). (End)

Maple

seq(binomial(n+8, 8)*5^n, n=0..18);

Mathematica

Table[Binomial[n + 8, 8] 5^n, {n, 0, 16}] (* or *)
CoefficientList[Series[1/(1 - 5 x)^9, {x, 0, 16}], x] (* Michael De Vlieger, Mar 20 2017 *)

Crossrefs

Sequence in context: A177728 A265615 A320363 * A268870 A387285 A049447

Adjacent sequences: A140343 A140344 A140345 * A140347 A140348 A140349

Keyword

nonn,easy

Author

Zerinvary Lajos, Jun 23 2008

Status

approved