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%I #31 Feb 26 2024 04:44:17
%S 0,1,7,141,5923,426457,46910271,7318002277,1536780478171,
%T 418004290062513,142957467201379447,60042136224579367741,
%U 30381320929637160076947,18228792557782296046168201,12796612375563171824410077103,10390849248957295521420982607637
%N a(n) = (2*n-2)*(2*n-1)*a(n-1)+1.
%H Harvey P. Dale, <a href="/A051397/b051397.txt">Table of n, a(n) for n = 0..225</a>
%H 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 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 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 a(n) = Sum_{k=0..n-1} (2*n-1)!/(2*k+1)!. a(n) = floor((2*n-1)!*sinh(1)). - _Vladeta Jovovic_, Aug 10 2002
%F Conjecture: a(n) +(-4*n^2+6*n-3)*a(n-1) +2*(2*n-3)*(n-2)*a(n-2)=0. - _R. J. Mathar_, Jan 31 2014
%F From _Peter Bala_, Sep 02 2016: (Start)
%F G.f. sinh(x)/(1 - x^2) = x + 7*x^3/3! + 141*x^5/5! + 5923*x^7/7! + ....
%F Mathar's conjectured recurrence a(n) = (4*n^2 - 6*n + 3)*a(n-1) - (2*n - 3)*(2*n - 4)*a(n-2) follows easily from the defining recurrence. The sequence b(n) := (2*n - 1)! also satisfies Mathar's recurrence but with b(1) = 1, b(2) = 6. This leads to the continued fraction representation a(n) = (2*n - 1)!*(1 + 1/(6 - 6/(21 - 20/(43 - ... - (2*n - 3)*(2*n - 4)/(4*n^2 - 6*n + 3) )))) for n >= 3. Taking the limit gives the continued fraction representation sinh(1) = A073742 = 1 + 1/(6 - 6/(21 - 20/(43 - ... - (2*n - 3)*(2*n - 4)/((4*n^2 - 6*n + 3) - ... )))). (End)
%t nxt[{n_,a_}]:={n+1,(2(n+1)-2)(2(n+1)-1)a+1}; Transpose[NestList[nxt,{0,0},20]][[2]] (* _Harvey P. Dale_, Jun 13 2016 *)
%Y Bisection of abs(A009628). Also bisection of A087208 and of A186763. Cf. A073742, A074790, A275651.
%K nonn,easy
%O 0,3
%A _Aleksandar Petojevic_
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