A140404 a(n) = binomial(n+5, 5)*7^n.

1, 42, 1029, 19208, 302526, 4235364, 54353838, 652246056, 7419298887, 80787921214, 848273172747, 8636963213424, 85649885199788, 830145041167176, 7886377891088172, 73606193650156272, 676256904160810749, 6126091955339109138, 54794489156088698401, 484498640959100070072

Offset

0,2

Comments

With a different offset, number of n-permutations of 8 objects:r,s,t,u,v,z,x,y with repetition allowed, containing exactly five (5) u's. Example: a(1)=42 because we have

uuuuur, uuuuru, uuuruu, uuruuu, uruuuu, ruuuuu

uuuuus, uuuusu, uuusuu, uusuuu, usuuuu, suuuuu,

uuuuut, uuuutu, uuutuu, uutuuu, utuuuu, tuuuuu,

uuuuuv, uuuuvu, uuuvuu, uuvuuu, uvuuuu, vuuuuu,

uuuuuz, uuuuzu, uuuzuu, uuzuuu, uzuuuu, zuuuuu,

uuuuux, uuuuxu, uuuxuu, uuxuuu, uxuuuu, xuuuuu,

uuuuuy, uuuuyu, uuuyuu, uuyuuu, uyuuuu, yuuuuu.

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..400

Index entries for linear recurrences with constant coefficients, signature (42,-735,6860,-36015,100842,-117649).

Formula

G.f.: 1/(1-7*x)^6. - Zerinvary Lajos, Aug 06 2008

a(n) = 42*a(n-1) - 735*a(n-2) + 6860*a(n-3) - 36015*a(n-4) + 100842*a(n-5) - 117649*a(n-6). - Harvey P. Dale, Sep 08 2011

From Amiram Eldar, Aug 28 2022: (Start)

Sum_{n>=0} 1/a(n) = 45360*log(7/6) - 27965/4.

Sum_{n>=0} (-1)^n/a(n) = 143360*log(8/7) - 229705/12. (End)

Maple

seq(binomial(n+5, 5)*7^n, n=0..17);

Mathematica

Table[Binomial[n+5, 5]7^n, {n, 0, 20}] (* or *) LinearRecurrence[ {42, -735, 6860, -36015, 100842, -117649}, {1, 42, 1029, 19208, 302526, 4235364}, 21] (* Harvey P. Dale, Sep 08 2011 *)

Prog

(Magma) [7^n* Binomial(n+5, 5): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011
(PARI) a(n)=binomial(n+5, 5)*7^n \\ Charles R Greathouse IV, Oct 07 2015

Crossrefs

Cf. A000389, A036084, A036219, A045543, A050982, A054849.

Sequence in context: A221050 A050988 A163741 * A075511 A016094 A004361

Adjacent sequences: A140401 A140402 A140403 * A140405 A140406 A140407

Keyword

nonn,easy

Author

Zerinvary Lajos, Jun 16 2008

Status

approved