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%I A085373 #24 May 14 2022 04:33:22
%S A085373 1,9,60,350,1890,9702,48048,231660,1093950,5080790,23279256,105462084,
%T A085373 473227300,2106121500,9307051200,40873466520,178520875830,
%U A085373 775924068150,3357800061000,14473885526100,62168784497820,266168518910580
%N A085373 a(n) = binomial(2n+1, n+1)*binomial(n+2, 2).
%H A085373 G. C. Greubel, <a href="/A085373/b085373.txt">Table of n, a(n) for n = 0..1000</a>
%F A085373 From _David Callan_, Nov 20 2003: (Start)
%F A085373 a(n-1) = Sum_{1<=i1<=i2<=...<=in<=n} (i1 + i2 + ... + in).
%F A085373 G.f.: (1 - x)/(1 - 4*x)^(5/2). (End)
%F A085373 a(n) = A119578(n+1)/2. - _Zerinvary Lajos_, Jun 19 2008
%F A085373 From _Amiram Eldar_, May 14 2022: (Start)
%F A085373 Sum_{n>=0} 1/a(n) = 4*Pi^2/9 - 4*Pi/sqrt(3) + 4.
%F A085373 Sum_{n>=0} (-1)^n/a(n) = 8*sqrt(5)*log(phi) - 16*log(phi)^2 - 4, where phi = A001622 is the golden ratio. (End)
%t A085373 Table[Binomial[2n+1, n+1]Binomial[n+2, 2], {n, 0, 30}]
%o A085373 (PARI) a(n)=binomial(2*n+1,n+1)*binomial(n+2,2)
%o A085373 (Magma) [Binomial(2*n+1, n+1)*Binomial(n+2, 2): n in [0..30]]; // _G. C. Greubel_, Feb 12 2019
%o A085373 (Sage) [binomial(2*n+1, n+1)*binomial(n+2, 2) for n in (0..30)] # _G. C. Greubel_, Feb 12 2019
%Y A085373 Cf. A000984, A002544, A119578.
%K A085373 easy,nonn
%O A085373 0,2
%A A085373 Mario Catalani (mario.catalani(AT)unito.it), Jun 26 2003

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