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a(n) = binomial(n+4, 4)*6^n.
4

%I #31 Aug 28 2022 04:20:04

%S 1,30,540,7560,90720,979776,9797760,92378880,831409920,7205552640,

%T 60526642176,495217981440,3961743851520,31084451758080,

%U 239794342133760,1822437000216576,13668277501624320,101306056776744960,742911083029463040,5395880497792942080

%N a(n) = binomial(n+4, 4)*6^n.

%C With a different offset, number of n-permutations (n=5) of 7 objects t, u, v, w, z, x, y with repetition allowed, containing exactly four (4) u's. Example: a(1)=30 because we have

%C uuuut, uuutu, uutuu, utuuu, tuuuu,

%C uuuuv, uuuvu, uuvuu, uvuuu, vuuuu,

%C uuuuw, uuuwu, uuwuu, uwuuu, wuuuu,

%C uuuuz, uuuzu, uuzuu, uzuuu, zuuuu,

%C uuuux, uuuxu, uuxuu, uxuuu, xuuuu,

%C uuuuy, uuuyu, uuyuu, uyuuu, yuuuu.

%H Vincenzo Librandi, <a href="/A139626/b139626.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (30,-360,2160,-6480,7776).

%F a(n) = A000332(n+4) * A000400(n). - _Michel Marcus_, Sep 11 2013

%F G.f.: 1 / (1-6*x)^5. - _Colin Barker_, Sep 25 2013

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

%F Sum_{n>=0} 1/a(n) = 548 - 3000*log(6/5).

%F Sum_{n>=0} (-1)^n/a(n) = 8232*log(7/6) - 1268. (End)

%p seq(binomial(n+4,4)*6^n,n=0..22);

%o (Magma) [6^n* Binomial(n+4, 4): n in [0..20]]; // _Vincenzo Librandi_, Oct 12 2011

%o (PARI) a(n)=binomial(n+4,4)*6^n \\ _Charles R Greathouse IV_, Sep 11 2013

%Y Cf. A000332, A000400.

%K nonn,easy

%O 0,2

%A _Zerinvary Lajos_, Jun 12 2008

%E More terms from _Colin Barker_, Sep 25 2013