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

Author

Zerinvary Lajos, Jun 23 2008

Keywords

• nonn
• easy

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

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

Programs

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 *)

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)