A139641 a(n) = binomial(n+4, 4)*7^n.

1, 35, 735, 12005, 168070, 2117682, 24706290, 271769190, 2853576495, 28852829005, 282757724249, 2699051004195, 25191142705820, 230595844768660, 2075362602917940, 18401548412539068, 161013548609716845, 1392293626213433895, 11911845468714934435, 100937216866479181265

Offset: 0

Author

Zerinvary Lajos, Jun 12 2008

Keywords

• nonn
• easy

Comments

With a different offset, number of n-permutations (n=5) of 8 objects s, t, u, v, w, z, x, y with repetition allowed, containing exactly four (4) u's. Example: a(1)=35 because we have

uuuus, uuusu, uusuu, usuuu, suuuu,

uuuut, uuutu, uutuu, utuuu, tuuuu,

uuuuv, uuuvu, uuvuu, uvuuu, vuuuu,

uuuuw, uuuwu, uuwuu, uwuuu, wuuuu,

uuuuz, uuuzu, uuzuu, uzuuu, zuuuu,

uuuux, uuuxu, uuxuu, uxuuu, xuuuu,

uuuuy, uuuyu, uuyuu, uyuuu, yuuuu.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..400
• Index entries for linear recurrences with constant coefficients, signature (35,-490,3430,-12005,16807).

Programs

Magma

[7^n* Binomial(n+4, 4): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

Maple

seq(binomial(n+4,4)*7^n,n=0..20);

Mathematica

Table[7^n * Binomial[n+4, 4], {n, 0, 20}] (* Amiram Eldar, Aug 28 2022 *)

Formula

From Amiram Eldar, Aug 28 2022: (Start)

Sum_{n>=0} 1/a(n) = 2800/3 - 6048*log(7/6).

Sum_{n>=0} (-1)^n/a(n) = 14336*log(8/7) - 5740/3. (End)