A141407 a(n) = binomial(n+7,7)*6^n.

1, 48, 1296, 25920, 427680, 6158592, 80061696, 960740352, 10808328960, 115288842240, 1175946190848, 11545653510144, 109683708346368, 1012465000120320, 9112185001082880, 80187228009529344, 691614841582190592, 5858384540460908544, 48819871170507571200, 400836836978904268800

Offset: 0

Author

Zerinvary Lajos, Aug 04 2008

Keywords

• nonn
• easy

Comments

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

If n=7 then a(0)= 1 because we have uuuuuuu

a(1)=48 because we have

uuuuuuus, uuuuuuut, uuuuuuuv, uuuuuuuz, uuuuuuux, uuuuuuuy,

uuuuuusu, uuuuuutu, uuuuuuvu, uuuuuuzu, uuuuuuxu, uuuuuuyu,

uuuuusuu, uuuuutuu, uuuuuvuu, uuuuuzuu, uuuuuxuu, uuuuuyuu,

uuuusuuu, uuuutuuu, uuuuvuuu, uuuuzuuu, uuuuxuuu, uuuuyuuu,

uuusuuuu, uuutuuuu, uuuvuuuu, uuuzuuuu, uuuxuuuu, uuuyuuuu,

uusuuuuu, uutuuuuu, uuvuuuuu, uuzuuuuu, uuxuuuuu, uuyuuuuu,

usuuuuuu, utuuuuuu, uvuuuuuu, uzuuuuuu, uxuuuuuu, uyuuuuuu,

suuuuuuu, tuuuuuuu, vuuuuuuu, zuuuuuuu, xuuuuuuu, yuuuuuuu.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (48,-1008,12096,-90720,435456,-1306368,2239488,-1679616).

Crossrefs

Cf. A038255.

Programs

Magma

[6^n* Binomial(n+7, 7): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Maple

seq(binomial(n+7,7)*6^n,n=0..16);

Mathematica

Table[Binomial[n + 7, 7] 6^n, {n, 0, 16}] (* Michael De Vlieger, Jul 24 2017 *)

PARI

vector(15,n,binomial(n+6,7)*6^(n-1)) \\ Derek Orr, Jul 24 2017

Formula

O.g.f.: 1/(1-6x)^8. - R. J. Mathar, Aug 08 2008

a(n) = 48*a(n-1) - 1008*a(n-2) + 12096*a(n-3) - 90720*a(n-4) + 435456*a(n-5) - 1306368*a(n-6) + 2239488*a(n-7) - 1679616*a(n-8) for n > 7. - Chai Wah Wu, Nov 12 2021

From Amiram Eldar, Aug 29 2022: (Start)

Sum_{n>=0} 1/a(n) = 656250*log(6/5) - 239295/2.

Sum_{n>=0} (-1)^n/a(n) = 4941258*log(7/6) - 7616973/10. (End)