A140406 - OEIS

%I #24 Aug 28 2022 04:22:19

%S 1,56,1792,43008,860160,15138816,242221056,3598712832,50381979648,

%T 671759728640,8598524526592,106309030510592,1275708366127104,

%U 14915974742409216,170468282770391040,1909244767028379648,21001692437312176128,227312435792084729856

%N a(n) = binomial(n+6, 6)*8^n.

%C With a different offset, number of n-permutations (n >= 6) of 9 objects: p, r, s, t, u, v, z, x, y with repetition allowed, containing exactly six (6) u's.

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

%C Example: a(1)=56 because we have

%C uuuuuup, uuuuupu, uuuupuu, uuupuuu, uupuuuu, upuuuuu, puuuuuu,

%C uuuuuur, uuuuuru, uuuuruu, uuuruuu, uuruuuu, uruuuuu, ruuuuuu,

%C uuuuuus, uuuuusu, uuuusuu, uuusuuu, uusuuuu, usuuuuu, suuuuuu,

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

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

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

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

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

%H Vincenzo Librandi, <a href="/A140406/b140406.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (56,-1344,17920,-143360,688128,-1835008,2097152).

%F G.f.: 1/(1-8*x)^7. - _Zerinvary Lajos_, Aug 06 2008

%F a(n) = 56*a(n-1) - 1344*a(n-2) + 17920*a(n-3) - 143360*a(n-4) + 688128*a(n-5) - 1835008*a(n-6) + 2097152*a(n-7). - _Harvey P. Dale_, Dec 15 2011

%F From _Amiram Eldar_, Aug 28 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 538628/5 - 806736*log(8/7).

%F Sum_{n>=0} (-1)^n/a(n) = 2834352*log(9/8) - 1669188/5. (End)

%p seq(binomial(n+6,6)*8^n,n=0..17);

%t Table[Binomial[n+6,6]8^n,{n,0,20}] (* or *) LinearRecurrence[ {56,-1344,17920,-143360,688128,-1835008,2097152},{1,56,1792,43008,860160,15138816,242221056},20] (* _Harvey P. Dale_, Dec 15 2011 *)

%o (Magma) [8^n* Binomial(n+6, 6): n in [0..20]]; // _Vincenzo Librandi_, Oct 16 2011

%o (PARI) a(n)=binomial(n+6,6)<<(3*n) \\ _Charles R Greathouse IV_, Dec 15 2011

%Y Cf. A000579, A002409, A036220, A036226, A050988, A054337, A140405.

%K nonn,easy

%O 0,2

%A _Zerinvary Lajos_, Jun 16 2008