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a(n) = binomial(n+9,9)*6^n.
2

%I #17 Aug 29 2022 04:41:40

%S 1,60,1980,47520,926640,15567552,233513280,3202467840,40831464960,

%T 489977579520,5585744406528,60935393525760,639821632020480,

%U 6496650417438720,64038411257610240,614768748073058304,5763457013184921600,52888193768049868800,475993743912448819200

%N a(n) = binomial(n+9,9)*6^n.

%C With a different offset, number of n-permutations (n>=9) of 7 objects: r, s, t, u, v, z, x, y with repetition allowed, containing exactly 9 u's.

%H Vincenzo Librandi, <a href="/A173123/b173123.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (60,-1620,25920,-272160,1959552,-9797760,33592320,-75582720,100776960,-60466176).

%F a(n) = C(n + 9, 9)*6^n.

%F From _Chai Wah Wu_, Nov 12 2021: (Start)

%F a(n) = 60*a(n-1) - 1620*a(n-2) + 25920*a(n-3) - 272160*a(n-4) + 1959552*a(n-5) - 9797760*a(n-6) + 33592320*a(n-7) - 75582720*a(n-8) + 100776960*a(n-9) - 60466176*a(n-10) for n > 9.

%F G.f.: 1/(6*x - 1)^10. (End)

%F From _Amiram Eldar_, Aug 29 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 21093750*log(6/5) - 107683641/28.

%F Sum_{n>=0} (-1)^n/a(n) = 311299254*log(7/6) - 959739813/20. (End)

%t Table[Binomial[n + 9, 9]*6^n, {n, 0, 20}]

%o (Magma) [6^n* Binomial(n+9, 9): n in [0..20]]; // _Vincenzo Librandi_, Oct 12 2011

%Y Cf. A081136, A081144, A139626, A036084, A050988, A141407, A172501.

%K nonn,easy

%O 0,2

%A _Zerinvary Lajos_, Feb 10 2010