A036222 - OEIS

%I #29 Sep 22 2022 03:58:30

%S 1,27,405,4455,40095,312741,2189187,14073345,84440070,478493730,

%T 2583866142,13389124554,66945622770,324428787270,1529449997130,

%U 7035469986798,31659614940591,139674771796725,605257344452475

%N Expansion of 1/(1-3*x)^9; 9-fold convolution of A000244 (powers of 3).

%C With a different offset, number of n-permutations (n>=8) of 4 objects: u, v, z, x with repetition allowed, containing exactly eight (8) u's. Example: a(1)=27 because we have uuuuuuuuv, uuuuuuuuz, uuuuuuuux, uuuuuuuvu, uuuuuuuzu, uuuuuuuxu, uuuuuuvuu, uuuuuuzuu, uuuuuuxuu, uuuuuvuuu, uuuuuzuuu, uuuuuxuuu, uuuuvuuuu, uuuuzuuuu, uuuuxuuuu, uuuvuuuuu, uuuzuuuuu, uuuxuuuuu, uuvuuuuuu, uuzuuuuuu, uuxuuuuuu, uvuuuuuuu, uzuuuuuuu, uxuuuuuuu, vuuuuuuuu, zuuuuuuuu, xuuuuuuuu. - _Zerinvary Lajos_, Jun 23 2008

%H Vincenzo Librandi, <a href="/A036222/b036222.txt">Table of n, a(n) for n = 0..400</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (27,-324,2268,-10206,30618,-61236,78732,-59049,19683).

%F a(n) = 3^n*binomial(n+8, 8).

%F a(n) = A027465(n+9, 9).

%F G.f.: 1/(1-3*x)^9.

%F a(0)=1, a(1)=27, a(2)=405, a(3)=4455, a(4)=40095, a(5)=312741, a(6)=2189187, a(7)=14073345, a(8)=84440070, a(n) = 27*a(n-1) - 324*a(n-2) + 2268*a(n-3) - 10206*a(n-4) + 30618*a(n-5) - 61236*a(n-6) + 78732*a(n-7) - 59049*a(n-8) + 19683*a(n-9). - _Harvey P. Dale_, Jan 07 2016

%F From _Amiram Eldar_, Sep 22 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 43632/35 - 3072*log(3/2).

%F Sum_{n>=0} (-1)^n/a(n) = 393216*log(4/3) - 3959208/35. (End)

%p seq(3^n*binomial(n+8,8), n=0..18); # _Zerinvary Lajos_, Jun 23 2008

%t Table[3^n*Binomial[n+8, 8], {n, 0, 20}] (* _Zerinvary Lajos_, Jan 31 2010 *)

%t CoefficientList[Series[1/(1-3x)^9,{x,0,30}],x] (* or *) LinearRecurrence[{27,-324, 2268,-10206,30618,-61236,78732,-59049,19683}, {1,27,405,4455,40095,312741, 2189187,14073345,84440070}, 30] (* _Harvey P. Dale_, Jan 07 2016 *)

%o (Sage) [3^n*binomial(n+8, 8) for n in range(30)] # _Zerinvary Lajos_, Mar 13 2009

%o (Magma) [3^n*Binomial(n+8, 8): n in [0..30]]; // _Vincenzo Librandi_, Oct 15 2011

%Y Cf. A027465.

%Y Sequences of the form 3^n*binomial(n+m, m): A000244 (m=0), A027471 (m=1), A027472 (m=2), A036216 (m=3), A036217 (m=4), A036219 (m=5), A036220 (m=6), A036221 (m=7), this sequence (m=8), A036223 (m=9), A172362 (m=10).

%K easy,nonn

%O 0,2

%A _Wolfdieter Lang_