A053135 - OEIS

%I #36 Mar 28 2024 23:55:32

%S 1,28,210,924,3003,8008,18564,38760,74613,134596,230230,376740,593775,

%T 906192,1344904,1947792,2760681,3838380,5245786,7059052,9366819,

%U 12271512,15890700,20358520,25827165,32468436,40475358,50063860,61474519,74974368,90858768

%N Binomial coefficients C(2*n+6,6).

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

%C Number of standard tableaux of shape (2n+1,1^6). - _Emeric Deutsch_, May 30 2004

%H Nathaniel Johnston, <a href="/A053135/b053135.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjić, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</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_07">Index entries for linear recurrences with constant coefficients</a>, signature (7,-21,35,-35,21,-7,1).

%F G.f.: (1 + 21*x + 35*x^2 + 7*x^3)/(1-x)^7.

%F a(n) = binomial(2*n+6, 6) = A000579(2*n+6).

%F a(n) = A000384(n+1)*A000384(n+2)*A000384(n+3)/90. - _Bruno Berselli_, Nov 12 2014

%F E.g.f.: (90 + 2430*x + 6975*x^2 + 5655*x^3 + 1710*x^4 + 204*x^5 + 8*x^6)* exp(x)/90. - _G. C. Greubel_, Sep 03 2018

%F From _Amiram Eldar_, Oct 21 2022: (Start)

%F Sum_{n>=0} 1/a(n) = 96*log(2) - 131/2.

%F Sum_{n>=0} (-1)^n/a(n) = 23/2 - 6*Pi + 12*log(2). (End)

%p seq(binomial(2*n+6,6),n=0..40); # _Nathaniel Johnston_, May 14 2011

%t Table[Binomial[2*n+6, 6], {n, 0, 30}] (* _G. C. Greubel_, Sep 03 2018 *)

%o (PARI) vector(30,n,n--; binomial(2*n+6, 6)) \\ _G. C. Greubel_, Sep 03 2018

%o (Magma) [Binomial(2*n+6, 6): n in [0..30]]; // _G. C. Greubel_, Sep 03 2018

%Y Cf. A000384, A000579, A002299, A007318, A053128, A053134, A190152.

%K nonn,easy

%O 0,2

%A _Wolfdieter Lang_