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 A262407 #17 Apr 09 2021 18:55:04 %S A262407 0,1,4,24,152,1010,6912,48328,343408,2471274,17966360,131717960, %T A262407 972488640,7223061040,53925450880,404400203280,3044645475296, %U A262407 23002424245754,174324246314184,1324800580881952,10093304926771600,77073430602848316,589761299099196224 %N A262407 a(n) = Sum_{k=0..n-1} C(n,k+1)*C(n,k)*C(n-1,k). %F A262407 a(n) = A000279(n)/(3*n) = (A000172(n)+4*A000172(n-1))*n/(3*(n+1)). %F A262407 a(n) ~ 8^n/(sqrt(3)*Pi*n) as n -> oo. %p A262407 a:= proc(n) option remember; `if`(n<2, n, %p A262407 ((21*n^3-49*n^2+30*n-8)*a(n-1)+ %p A262407 (8*(n-1))*(n-2)*(3*n-1)*a(n-2))/ %p A262407 ((3*n-4)*(n+1)*(n-1))) %p A262407 end: %p A262407 seq(a(n), n=0..30); # _Alois P. Heinz_, Sep 22 2015 %t A262407 f[n_]:=HypergeometricPFQ[{-n, -n, -n}, {1, 1}, -1]; a[n_]:=n^2 (f[n] + 4 f[n - 1])/(3 n^2 + 3 n); Array[a, 25] (* _Vincenzo Librandi_, Sep 22 2015 *) %t A262407 Table[Sum[Binomial[n,k+1]Binomial[n,k]Binomial[n-1,k],{k,0,n-1}],{n,0,30}] (* _Harvey P. Dale_, Apr 09 2021 *) %o A262407 (PARI) a(n)=sum(k=0, n-1, binomial(n, k+1)*binomial(n, k)*binomial(n-1, k)) %Y A262407 Cf. A000489, A000535, A000279. %K A262407 nonn %O A262407 0,3 %A A262407 _M. F. Hasler_, Sep 21 2015