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 A051396 #39 Feb 26 2024 04:40:54
%S A051396 0,1,3,37,1111,62217,5599531,739138093,134523132927,32285551902481,
%T A051396 9879378882159187,3754163975220491061,1734423756551866870183,
%U A051396 957401913616630512341017,622311243850809833021661051,470467300351212233764375754557,409306551305554643375006906464591
%N A051396 a(n) = (2*n-2)*(2*n-3)*a(n-1)+1.
%C A051396 The sequence 1,0,3,0,37,... has e.g.f. cosh(x)/(1-x^2) with a(n) = Sum_{k=0..n} C(n,k)k!(1+(-1)^k)(1+(-1)^(n-k))/4. - _Paul Barry_, May 01 2005
%H A051396 Seiichi Manyama, <a href="/A051396/b051396.txt">Table of n, a(n) for n = 0..225</a>
%H A051396 Romeo Mestrovic, <a href="http://arxiv.org/abs/1312.7037">Variations of Kurepa's left factorial hypothesis</a>, arXiv preprint arXiv:1312.7037 [math.NT], 2013.
%H A051396 Romeo Mestrovic, <a href="https://doi.org/10.2298/FIL1510207M">The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis</a>, Filomat 29:10 (2015), 2207-2215; DOI 10.2298/FIL1510207M.
%H A051396 Aleksandar Petojevic, <a href="http://elib.mi.sanu.ac.rs/files/journals/flmt/12/flmn12p29-37.pdf">On Kurepa's Hypothesis for the Left Factorial</a>, FILOMAT (Nis), 12:1 (1998), p. 29-37.
%F A051396 a(n) = Sum_{k=0..n-1} (2*n-2)!/(2*k)! = floor((2*n-2)!*cosh(1)), n>=1. - _Vladeta Jovovic_, Aug 10 2002
%F A051396 a(n+1) = Sum_{k=0..2n}, C(2n, k)*k!*(1+(-1)^k)^2. - _Paul Barry_, May 01 2005
%F A051396 a(n) +(-4*n^2+10*n-7)*a(n-1) +2*(n-2)*(2*n-5)*a(n-2)=0. - _R. J. Mathar_, Nov 26 2012
%F A051396 From _Peter Bala_, Sep 05 2016: (Start)
%F A051396 The sequence b(n) := (2*n - 2)! also satisfies Mathar's recurrence with b(1) = 1, b(2) = 2. This leads to the continued fraction representation a(n) = (2*n - 2)!*(1 + 1/(2 - 2/(13 - 12/(31 - ... - (2*n - 4)*(2*n - 5)/(4*n^2 - 10*n + 7) )))) for n >= 3. Taking the limit gives the continued fraction representation cosh(1) = A073743 = 1 + 1/(2 - 2/(13 - 12/(31 - ... - (2*n - 4)*(2*n - 5)/((4*n^2 - 10*n + 7) - ... )))). (End)
%p A051396 A051396 := proc(n) option remember; if n <= 1 then n else (2*n-2)*(2*n-3)*A051396(n-1)+1; fi; end;
%t A051396 a[0] = 0; a[n_] := a[n] = (2*n-2)*(2*n-3)*a[n-1] + 1;
%t A051396 Table[a[n], {n, 0, 16}] (* _Jean-François Alcover_, Dec 11 2017 *)
%t A051396 nxt[{n_,a_}]:={n+1,a(4n^2-2n)+1}; NestList[nxt,{0,0},20][[;;,2]] (* _Harvey P. Dale_, Sep 26 2023 *)
%Y A051396 Bisection of abs(A009179(n)). Cf. A049470 (cos(1)), A073743 (cosh(1)), A275651.
%K A051396 nonn,easy
%O A051396 0,3
%A A051396 _Aleksandar Petojevic_