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%I #38 Sep 08 2022 08:45:37
%S 1,3,37,743,20841,751019,33065677,1720166223,103243039057,
%T 7022246822099,533794001518581,44845718374382903,4126339884444745657,
%U 412678834162848948603,44573440429472131194781,5170931768652930067543199,641240112753392800506551457,84648865815216502596932335523
%N Main diagonal of A143410.
%C The sequence of convergents of the continued fraction expansion sqrt(e) = 1 + 2/(3 + 1/(12 + 1/(20 + 1/(28 + 1/(36 + ... ))))) begins [1/1, 5/3, 61/37, 1225/743, ...]. The partial denominators are this sequence; the numerators are A065919. - _Peter Bala_, Jan 02 2020
%H Robert Israel, <a href="/A143412/b143412.txt">Table of n, a(n) for n = 0..333</a>
%F a(n) = (-1)^n*Sum_{k = 0..n} (-2)^k*(n+k)!/((n-k)!*k!) = (-1)^n*y_n(-4), where y_n(x) denotes the n-th Bessel polynomial.
%F Recurrence relation: a(0) = 1, a(1) = 3, a(n) = 4*(2*n-1)*a(n-1) + a(n-2) for n >= 2. Sequence A065919 satisfies the same recurrence relation.
%F sqrt(e) = 1 + 2*Sum_{n >= 0} (-1)^n/(a(n)*a(n+1)) = 1 + 2*(1/(1*3) - 1/(3*37) + 1/(37*743) - ...) (see A019774).
%F G.f.: 1/Q(0), where Q(k)= 1 + x - 4*x*(k+1)/Q(k+1); (continued fraction). - _Sergei N. Gladkovskii_, May 17 2013
%F a(n) = (-1)^n * hypergeom([-n,n+1],[],2). - _Robert Israel_, Jan 03 2016
%F a(n) ~ 2^(3*n + 1/2) * n^n / exp(n + 1/4). - _Vaclav Kotesovec_, Jan 02 2020
%F a(n) is the expectation of U_{2n}(X) where X is a standard Gaussian random variable and U_n is the n-th Chebyshev polynomial of second kind (conjectured). - _Benjamin Dadoun_, Dec 16 2020
%F a(n) = 2^n*KummerU(-n, -2*n, -1/2). - _Peter Luschny_, May 10 2022
%p a := n -> (-1)^n*add ((-2)^k*(n+k)!/((n-k)!*k!),k = 0..n): seq(a(n),n = 0..16);
%p seq(simplify(2^n*KummerU(-n,-2*n,-1/2)), n=0..17); # _Peter Luschny_, May 10 2022
%t RecurrenceTable[{ a[n + 2] == 4*(2 n + 3)*a[n + 1] + a[n], a[0] == 1, a[1] == 3}, a, {n, 0, 20}] (* _G. C. Greubel_, Jan 03 2016 *)
%o (PARI) a(n) = (-1)^n*sum(k=0,n, (-2)^k*(n+k)!/((n-k)!*k!) ); \\ _Joerg Arndt_, May 17 2013
%o (Magma) I:=[1,3]; [n le 2 select I[n] else 4*(2*n -3)*Self(n - 1) + Self(n-2): n in [1..20]]; // _Vincenzo Librandi_, Jan 03 2016
%Y Cf. A065919, A143410, A019774.
%K easy,nonn
%O 0,2
%A _Peter Bala_, Aug 14 2008
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