A143412 Main diagonal of A143410.

1, 3, 37, 743, 20841, 751019, 33065677, 1720166223, 103243039057, 7022246822099, 533794001518581, 44845718374382903, 4126339884444745657, 412678834162848948603, 44573440429472131194781, 5170931768652930067543199, 641240112753392800506551457, 84648865815216502596932335523

Offset: 0

Author

Peter Bala, Aug 14 2008

Keywords

• easy
• nonn

Comments

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

Links

• Robert Israel, Table of n, a(n) for n = 0..333

Crossrefs

Cf. A065919, A143410, A019774.

Programs

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

Maple

a := n -> (-1)^n*add ((-2)^k*(n+k)!/((n-k)!*k!),k = 0..n): seq(a(n),n = 0..16);
seq(simplify(2^n*KummerU(-n,-2*n,-1/2)), n=0..17); # Peter Luschny, May 10 2022

Mathematica

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 *)

PARI

a(n) = (-1)^n*sum(k=0,n, (-2)^k*(n+k)!/((n-k)!*k!) ); \\ Joerg Arndt, May 17 2013

Formula

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.

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.

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).

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

a(n) = (-1)^n * hypergeom([-n,n+1],[],2). - Robert Israel, Jan 03 2016

a(n) ~ 2^(3*n + 1/2) * n^n / exp(n + 1/4). - Vaclav Kotesovec, Jan 02 2020

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

a(n) = 2^n*KummerU(-n, -2*n, -1/2). - Peter Luschny, May 10 2022