A140405 a(n) = binomial(n+6, 6)*5^n.

1, 35, 700, 10500, 131250, 1443750, 14437500, 134062500, 1173046875, 9775390625, 78203125000, 604296875000, 4532226562500, 33120117187500, 236572265625000, 1656005859375000, 11385040283203125, 77016448974609375, 513442993164062500, 3377914428710937500

Offset: 0

Author

Zerinvary Lajos, Jun 16 2008

Keywords

• nonn
• easy

Comments

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

If n=6 then a(0)=1.

Example: a(1)=35 because we have

uuuuuut, uuuuutu, uuuutuu, uuutuuu, uutuuuu, utuuuuu, tuuuuuu,

uuuuuuv, uuuuuvu, uuuuvuu, uuuvuuu, uuvuuuu, uvuuuuu, vuuuuuu,

uuuuuuz, uuuuuzu, uuuuzuu, uuuzuuu, uuzuuuu, uzuuuuu, zuuuuuu,

uuuuuux, uuuuuxu, uuuuxuu, uuuxuuu, uuxuuuu, uxuuuuu, xuuuuuu,

uuuuuuy, uuuuuyu, uuuuyuu, uuuyuuu, uuyuuuu, uyuuuuu, yuuuuuu.

Links

• Index entries for linear recurrences with constant coefficients, signature (35,-525,4375,-21875,65625,-109375,78125).

Crossrefs

Cf. A000579, A002409, A036220, A036226, A050988, A054337, A140406.

Programs

Maple

seq(binomial(n+6,6)*5^n,n=0..18);

Mathematica

Table[Binomial[n+6,6]5^n,{n,0,20}] (* Harvey P. Dale, Dec 03 2017 *)

Formula

G.f.: 1/(1-5*x)^7. - Zerinvary Lajos, Aug 06 2008

From Amiram Eldar, Aug 29 2022: (Start)

Sum_{n>=0} 1/a(n) = 6856 - 30720*log(5/4).

Sum_{n>=0} (-1)^n/a(n) = 233280*log(6/5) - 42531. (End)