A053137 - OEIS

%I #33 Mar 29 2024 02:12:59

%S 1,45,495,3003,12870,43758,125970,319770,735471,1562275,3108105,

%T 5852925,10518300,18156204,30260340,48903492,76904685,118030185,

%U 177232627,260932815,377348994,536878650,752538150,1040465790,1420494075,1916797311,2558620845,3381098545

%N Binomial coefficients C(2*n+8,8).

%C Even-indexed members of ninth column of Pascal's triangle A007318.

%C Number of standard tableaux of shape (2n+1,1^8). - _Emeric Deutsch_, May 30 2004

%H Vincenzo Librandi, <a href="/A053137/b053137.txt">Table of n, a(n) for n = 0..200</a>

%H Milan Janjić, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfor.pdf">Two Enumerative Functions</a>.

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (9,-36,84,-126,126,-84,36,-9,1).

%F a(n) = binomial(2*n+8, 8) = A000581(2*n+8).

%F G.f.: (1+36*x+126*x^2+84*x^3+9*x^4) / (1-x)^9 = (1+3*x) * (3*x^3+27*x^2+33*x+1) / (1-x)^9.

%F From _Amiram Eldar_, Nov 03 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 512*log(2) - 5308/15.

%F Sum_{n>=0} (-1)^n/a(n) = 16*Pi + 32*log(2) - 1072/15. (End)

%t Table[Binomial[2*n+8, 8], {n, 0, 30}] (* _G. C. Greubel_, Sep 03 2018 *)

%t LinearRecurrence[{9,-36,84,-126,126,-84,36,-9,1},{1,45,495,3003,12870,43758,125970,319770,735471},30] (* _Harvey P. Dale_, Jul 02 2022 *)

%o (Magma) [Binomial(2*n+8,8): n in [0..30]]; // _Vincenzo Librandi_, Oct 07 2011

%o (PARI) a(n)=binomial(2*n+8,8) \\ _Charles R Greathouse IV_, Oct 07 2015

%Y Cf. A007318, A053136, A000581, A053130.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_