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 A319578 #16 Aug 25 2025 10:12:52
%S A319578 1,10,140,2310,42042,816816,16628040,350574510,7595781050,
%T A319578 168212023980,3792416540640,86787993910800,2011383287449200,
%U A319578 47123837020238400,1114478745528638160,26575401262863040830,638330716607984804250,15431925043610580004500,375239440534109892741000
%N A319578 a(n) = (1/3)*(n+2)^2*(3*n+3)!/(n+2)!^3.
%C A319578 Number of Schröder paths of length 2n+1 having n peaks.
%F A319578 a(n) = (n+2)*(3*n+2)!/((n+2)!^2*n!).
%F A319578 a(n) = A060693(2n+1,n).
%F A319578 G.f.: (hypergeom([1/3, 2/3], [2], 27*x) - 1)/(3*x). - _Stefano Spezia_, Aug 25 2025
%p A319578 a := n -> (n+2)*(3*n+2)!/((n+2)!^2*n!): seq(a(n), n = 0..18);
%t A319578 Table[(n+2) (3*n+2)! / ((n+2)!^2 n!), {n, 0, 30}] (* _Vincenzo Librandi_, Oct 01 2018 *)
%o A319578 (PARI) a(n) = (1/3)*(n+2)^2*(3*n+3)!/(n+2)!^3; \\ _Michel Marcus_, Oct 01 2018
%o A319578 (Magma) [(1/3)*(n+2)^2*Factorial(3*n+3)/Factorial(n+2)^3: n in [0..20]]; // _Vincenzo Librandi_, Oct 01 2018
%Y A319578 Cf. A007004, A060693, A215287.
%K A319578 nonn,changed
%O A319578 0,2
%A A319578 _Peter Luschny_, Sep 30 2018