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%I #34 Mar 29 2024 02:13:09
%S 1,55,715,5005,24310,92378,293930,817190,2042975,4686825,10015005,
%T 20160075,38567100,70607460,124403620,211915132,350343565,563921995,
%U 886163135,1362649145,2054455634,3042312350,4431613550,6358402050,8996462475,12565671261,17341763505
%N Binomial coefficients C(2*n+9,9).
%C Even-indexed members of tenth column of Pascal's triangle A007318.
%C Number of standard tableaux of shape (2n+1,1^9). - _Emeric Deutsch_, May 30 2004
%H Vincenzo Librandi, <a href="/A053138/b053138.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_10">Index entries for linear recurrences with constant coefficients</a>, signature (10,-45,120,-210,252,-210,120,-45, 10,-1).
%F a(n) = binomial(2*n+9, 9) = A000582(2*n+9).
%F G.f.: (1 + 45*x + 210*x^2 + 210*x^3 + 45*x^4 + x^5) / (1-x)^10.
%F G.f.: (1 + x) * (x^4 + 44*x^3 + 166*x^2 + 44*x + 1) / (1-x)^10.
%F a(n) = 10*a(n-1) - 45*a(n-2) + 120*a(n-3) - 210*a(n-4) + 252*a(n-5) - 210*a(n-6) + 120*a(n-7) - 45*a(n-8) + 10*a(n-9) - a(n-10) for n > 9. - _Wesley Ivan Hurt_, Dec 05 2016
%F From _Amiram Eldar_, Nov 03 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 1152*log(2) - 27912/35.
%F Sum_{n>=0} (-1)^n/a(n) = 36*Pi - 3924/35. (End)
%p A053138:=n->binomial(2*n+9,9): seq(A053138(n), n=0..40); # _Wesley Ivan Hurt_, Dec 05 2016
%t Table[Binomial[2n+9,9],{n,0,30}] (* _Harvey P. Dale_, Dec 08 2011 *)
%o (Magma) [Binomial(2*n+9,9): n in [0..30]]; // _Vincenzo Librandi_, Oct 07 2011
%o (PARI) for(n=0,30, binomial(2*n+9, 9), ", ")) \\ _G. C. Greubel_, Sep 03 2018
%Y Cf. A000582, A007318, A053131, A053137.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_
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