A143412 Main diagonal of A143410.

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

Offset

0,2

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

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

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 *)

Prog

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

Crossrefs

Cf. A065919, A143410, A019774.

Sequence in context: A245265 A362672 A143639 * A331656 A003717 A354020

Adjacent sequences: A143409 A143410 A143411 * A143413 A143414 A143415

Keyword

easy,nonn

Author

Peter Bala, Aug 14 2008

Status

approved