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%I #33 Mar 28 2024 23:55:45
%S 1,36,330,1716,6435,19448,50388,116280,245157,480700,888030,1560780,
%T 2629575,4272048,6724520,10295472,15380937,22481940,32224114,45379620,
%U 62891499,85900584,115775100,154143080,202927725,264385836,341149446,436270780,553270671,696190560
%N Binomial coefficients C(2*n+7,7).
%C Even-indexed members of eighth column of Pascal's triangle A007318.
%C Number of standard tableaux of shape (2n+1,1^7). - _Emeric Deutsch_, May 30 2004
%H Vincenzo Librandi, <a href="/A053136/b053136.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_08">Index entries for linear recurrences with constant coefficients</a>, signature (8,-28,56,-70,56,-28,8,-1).
%F a(n) = binomial(2*n+7, 7) = A000580(2*n+7).
%F G.f.: (1 + 28*x + 70*x^2 + 28*x^3 + x^4)/(1-x)^8.
%F E.g.f.: (630 + 22050*x + 81585*x^2 + 87465*x^3 + 36960*x^4 + 6888*x^5 + 560*x^6 + 16*x^7)*exp(x)/630. - _G. C. Greubel_, Sep 03 2018
%F From _Amiram Eldar_, Nov 03 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 224*log(2) - 4627/30.
%F Sum_{n>=0} (-1)^n/a(n) = 28*log(2) - 553/30. (End)
%t Table[Binomial[2*n+7, 7], {n, 0, 30}] (* _G. C. Greubel_, Sep 03 2018 *)
%o (Magma) [Binomial(2*n+7,7): n in [0..30]]; // _Vincenzo Librandi_, Oct 07 2011
%o (PARI) a(n)=binomial(2*n+7,7) \\ _Charles R Greathouse IV_, Oct 07 2015
%Y Cf. A007318, A053135, A000580, A053129.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_
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