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%I #46 Oct 21 2022 09:44:54
%S 1,15,70,210,495,1001,1820,3060,4845,7315,10626,14950,20475,27405,
%T 35960,46376,58905,73815,91390,111930,135751,163185,194580,230300,
%U 270725,316251,367290,424270,487635,557845,635376,720720,814385,916895,1028790,1150626,1282975
%N Binomial coefficients C(2*n+4,4).
%C Even-indexed members of fifth column of Pascal's triangle A007318.
%C Number of standard tableaux of shape (2n+1,1^4). - _Emeric Deutsch_, May 30 2004
%C Number of integer solutions to -n <= x <= y <= z <= w <= n. - _Michael Somos_, Dec 28 2011
%H Vincenzo Librandi, <a href="/A053134/b053134.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_05">Index entries for linear recurrences with constant coefficients</a>, signature (5,-10,10,-5,1).
%F a(n) = binomial(2*n+4, 4) = A000332(2*n+4).
%F G.f.: (1 + 10*x + 5*x^2)/(1-x)^5.
%F a(1 - n) = A053126(n). - _Michael Somos_, Dec 28 2011
%F E.g.f.: (6 + 84*x + 123*x^2 + 44*x^3 + 4*x^4)*exp(x)/6. - _G. C. Greubel_, Sep 03 2018
%F a(n) = (1/6)*(n + 1)*(n + 2)*(2*n + 1)*(2*n + 3). - _Gerry Martens_, Oct 13 2022
%F From _Amiram Eldar_, Oct 21 2022: (Start)
%F Sum_{n>=0} 1/a(n) = 16*log(2) - 10.
%F Sum_{n>=0} (-1)^n/a(n) = 10 - 2*Pi - 4*log(2). (End)
%e 1 + 15*x + 70*x^2 + 210*x^3 + 495*x^4 + 1001*x^5 + 1820*x^6 + 3060*x^7 + ...
%t Table[Binomial[2*n+4,4], {n,0,30}] (* or *) LinearRecurrence[{5,-10,10,-5 ,1}, {1, 15, 70, 210, 495}, 30] (* _G. C. Greubel_, Sep 03 2018 *)
%o (Magma) [Binomial(2*n+4,4): n in [0..30]]; // _Vincenzo Librandi_, Oct 07 2011
%o (PARI) for(n=0,30, print1(binomial(2*n+4,4), ", ")) \\ _G. C. Greubel_, Sep 03 2018
%Y Cf. A000447, A002299, A053126, A000332.
%K nonn,easy
%O 0,2
%A _Wolfdieter Lang_
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