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 A006551 M3426 #48 Dec 20 2024 11:07:17
%S A006551 1,1,4,11,66,302,2416,15619,156190,1310354,15724248,162512286,
%T A006551 2275172004,27971176092,447538817472,6382798925475,114890380658550,
%U A006551 1865385657780650,37307713155613000,679562217794156938,14950368791471452636
%N A006551 Maximal Eulerian numbers.
%C A006551 From _Peter Luschny_, Aug 08 2010: (Start)
%C A006551 Define A(n,k) as the number of permutations of {1,2,..,n} with k ascents.
%C A006551 A(n,k) = sum_{j=0}^k (-1)^j binomial(n+1,j)(k-j+1)^n.
%C A006551 Then a(n) = A(n, floor(n/2)). The Digital Library of Mathematical Functions calls the A(n,k) Eulerian numbers. With this terminology a(n) are the middle Eulerian numbers and A180056 the central Eulerian numbers. (End)
%C A006551 Number of permutations of {1,2,..,n} with floor(n/2) descents. - _Joerg Arndt_, Aug 15 2014
%D A006551 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 243.
%D A006551 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H A006551 Alois P. Heinz, <a href="/A006551/b006551.txt">Table of n, a(n) for n = 1..450</a>
%H A006551 Digital Library of Mathematical Functions, <a href="http://dlmf.nist.gov/26.14#T1">Table 26.14.1</a> [_Peter Luschny_, Aug 08 2010]
%H A006551 Herman Chau, <a href="https://arxiv.org/abs/2412.10532">On Enumerating Higher Bruhat Orders Through Deletion and Contraction</a>, arXiv:2412.10532 [math.CO], 2024. See p. 20.
%H A006551 L. Lesieur and J.-N. Nicolas, <a href="http://dx.doi.org/10.1016/S0195-6698(05)80018-6">On the Eulerian numbers M_n = max_{1<=k<=n} A(n,k)</a>, European J. Combin., 13 (1992), 379-399.
%H A006551 Robert G. Wilson v, <a href="/A007347/a007347.pdf">Letter to N. J. A. Sloane, Apr. 1994</a>
%F A006551 a(n) = sum_{0<=j<=floor(n/2)} (-1)^j binomial(n+1,j) (floor(n/2)-j+1)^n. [_Peter Luschny_, Aug 08 2010]
%F A006551 a(n+1)/a(n) ~ n. - _Ran Pan_, Oct 26 2015
%F A006551 a(n) ~ 2 * sqrt(3) * n^n / exp(n). - _Vaclav Kotesovec_, Oct 28 2021
%p A006551 a := proc(n) local j,k; k := iquo(n,2);
%p A006551 add((-1)^j*binomial(n+1,j)*(k-j+1)^n,j=0..k) end:
%p A006551 # _Peter Luschny_, Aug 08 2010
%p A006551 # Computation by recursion:
%p A006551 A006551 := proc(r) local W; W := proc(m) local A,n,k;
%p A006551 A:=[seq(1, n=1..m)]; if m < 2 then RETURN(1) fi;
%p A006551 for n from 2 to m-1 do for k from 2 to m do
%p A006551 A[k] := n*A[k-1]+k*A[k] od od; [A[m-1],A[m]] end:
%p A006551 W((r+2+irem(r,2))/2)[2-irem(r,2)] end:
%p A006551 # _Peter Luschny_, Jan 12 2011
%t A006551 a[n_] := With[{k = Quotient[n, 2]}, Sum[(-1)^j*Binomial[n+1, j]*(k-j+1)^n, {j, 0, k}]]; Array[a, 25] (* _Jean-François Alcover_, Feb 19 2017, after _Peter Luschny_ *)
%Y A006551 Cf. A008292. Bisections are A025585 and A180056.
%K A006551 nonn
%O A006551 1,3
%A A006551 _N. J. A. Sloane_, _Mira Bernstein_, _Robert G. Wilson v_