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

Author

Zerinvary Lajos, Jun 16 2008

Keywords

• nonn
• easy

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).

Crossrefs

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

Programs

Magma

[7^n* Binomial(n+5, 5): n in [0..20]]; // Vincenzo Librandi, Oct 12 2011

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 *)

PARI

a(n)=binomial(n+5,5)*7^n \\ Charles R Greathouse IV, Oct 07 2015

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)