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 A129833 #27 Mar 10 2021 03:20:30
%S A129833 1,3,11,52,309,2221,18703,180216,1952457,23466223,309577971,
%T A129833 4444537868,68948023741,1148825560377,20455144724407,387479309532976,
%U A129833 7778881684953873,164942847995071611,3682885668837002587,86359724102207331876,2121535102985378053061,54482075844410029721893,1459677302947807284662751
%N A129833 a(n) = Sum_{k = 0..n } binomial(n + 1, k + 1)*binomial(n, k)*k!.
%H A129833 Seiichi Manyama, <a href="/A129833/b129833.txt">Table of n, a(n) for n = 0..443</a>
%H A129833 F. Hivert, J.-C. Novelli and J.-Y. Thibon, <a href="https://arxiv.org/abs/math/0605262">Commutative combinatorial Hopf algebras</a>, arXiv:math/0605262 [math.CO], 2006.
%F A129833 Conjecture: +(n+1)*a(n) -n*(2*n+5)*a(n-1) +(n-1)*(n^2+6*n+3)*a(n-2) -(n-2)*(3*n^2-2)*a(n-3) +(n-2)*(n-3)*(3*n-4)*a(n-4) -(n-4)*(n-3)^2*a(n-5) = 0. - _R. J. Mathar_, Feb 28 2015
%F A129833 Conjecture: (n+1)*(n^2-4*n+2)*a(n) -n*(2*n^3-5*n^2-6*n+3)*a(n-1) +n*(n-1)*(n^3-2*n^2-2*n-2)*a(n-2) -(n-2)*(n^2-2*n-1)*(n-1)^2*a(n-3) = 0. - _R. J. Mathar_, Feb 28 2015
%F A129833 a(n) ~ exp(2*sqrt(n) - n - 1/2) * n^(n + 1/4) / sqrt(2) * (1 + 79/(48*sqrt(n))). - _Vaclav Kotesovec_, Oct 12 2016
%F A129833 From _G. C. Greubel_, Mar 10 2021: (Start)
%F A129833 a(n) = Sum_{k=0..n} binomial(n,k)^2 * ((n+1)*k!/(k+1)).
%F A129833 a(n) = (n+1)*Hypergeometric3F1([-n, -n, 1], [2], 1). (End)
%p A129833 A129833 := proc(n)
%p A129833 add(A176120(n,k),k=0..n) ;
%p A129833 end proc: # _R. J. Mathar_, Feb 28 2015
%t A129833 a[n_]:= Sum[Binomial[n+1, k+1]*Binomial[n, k]*k!, {k,0,n}]; Table[a[n], {n,0,30}]
%o A129833 (Sage) [sum( binomial(n,k)^2*((n+1)*factorial(k)/(k+1)) for k in (0..n)) for n in (0..30)] # _G. C. Greubel_, Mar 10 2021
%o A129833 (Magma) [(&+[Binomial(n,k)^2*((n+1)*Factorial(k)/(k+1)): k in [0..n]]): n in [0..30]]; // _G. C. Greubel_, Mar 10 2021
%o A129833 (PARI) a(n) = sum(k= 0, n, binomial(n+1, k+1)*binomial(n, k)*k!); \\ _Michel Marcus_, Mar 10 2021
%Y A129833 Cf. A052852, A176120.
%K A129833 nonn,easy
%O A129833 0,2
%A A129833 _Roger L. Bagula_, May 21 2007
%E A129833 Edited by _N. J. A. Sloane_, Sep 30 2007