This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A085374 #25 Sep 06 2025 15:44:35
%S A085374 1,12,100,700,4410,25872,144144,772200,4011150,20323160,100876776,
%T A085374 492156392,2366136500,11232648000,52739956800,245240799120,
%U A085374 1130632213590,5172827121000,23504600427000,106141827191400,476627347816620,2129348151284640,9468445336740000
%N A085374 a(n) = binomial(2n+1, n+1)*binomial(n+3, 3).
%F A085374 a(n) = A000292(n+1)*A000984(n+1)/2. - _Zerinvary Lajos_, Jan 18 2007, corrected Aug 09 2015
%F A085374 From _R. J. Mathar_, Aug 09 2015: (Start)
%F A085374 D-finite with recurrence n*(n+1)*a(n) - 2*(n+3)*(2*n+1)*a(n-1) = 0.
%F A085374 G.f.: 2F1(3/2,4;2;4x). (End)
%F A085374 a(n) ~ 2^(2*n)*n^(5/2)/(3*sqrt(Pi)). - _Stefano Spezia_, Aug 31 2025
%F A085374 From _Amiram Eldar_, Sep 06 2025: (Start)
%F A085374 a(n) = A001700(n) * A000292(n+1).
%F A085374 Sum_{n>=0} 1/a(n) = 10*sqrt(3)*Pi - 8*Pi^2/3 - 27.
%F A085374 Sum_{n>=0} (-1)^n/a(n) = 84*sqrt(5)*log(phi) - 192*log(phi)^2 - 45, where phi is the golden ratio (A001622). (End)
%p A085374 seq(binomial(n+2,3)/2*binomial(2*n,n), n=1..20); # _Zerinvary Lajos_, Jan 18 2007
%t A085374 Table[Binomial[2 n + 1, n + 1]Binomial[n + 3, 3], {n, 0, 30}]
%Y A085374 Cf. A001700, A002457, A085373.
%Y A085374 Cf. A000292, A000984, A001622.
%K A085374 easy,nonn,changed
%O A085374 0,2
%A A085374 Mario Catalani (mario.catalani(AT)unito.it), Jun 26 2003