login
a(n) = binomial(9*n, n).
12

%I #28 Sep 08 2022 08:45:49

%S 1,9,153,2925,58905,1221759,25827165,553270671,11969016345,

%T 260887834350,5720645481903,126050526132804,2788629694000605,

%U 61902409203193230,1378095785451705375,30756373941461374800,687917389635036844569,15415916972482007401455,346051021610256116115150

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

%H Vincenzo Librandi, <a href="/A169958/b169958.txt">Table of n, a(n) for n = 0..110</a>

%F a(n) = C(9*n-1, n-1)*C(81*n^2, 2)/(3*n*C(9*n+1, 3)), n > 0. - _Gary Detlefs_, Jan 02 2014

%F From _Peter Bala_, Feb 21 2022: (Start)

%F The o.g.f. A(x) is algebraic: (1 - A(x))*(1 + 8*A(x))^8 + (9^9)*x*A(x)^9 = 0.

%F Sum_{n >= 1} a(n)*( x*(8*x + 9)^8/(9^9*(1 + x)^9) )^n = x. (End)

%o (Magma) [Binomial(9*n, n): n in [0..50] ]; // _Vincenzo Librandi_, Apr 21 2011

%Y binomial(k*n,n): A000984 (k = 2), A005809 (k = 3), A005810 (k = 4), A001449 (k = 5), A004355 (k = 6), A004368 (k = 7), A004381 (k = 8), A169959 - A169961 (k = 10 thru 12).

%K nonn

%O 0,2

%A _N. J. A. Sloane_, Aug 07 2010