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 A143412 Main diagonal of A143410. 3
 1, 3, 37, 743, 20841, 751019, 33065677, 1720166223, 103243039057, 7022246822099, 533794001518581, 44845718374382903, 4126339884444745657, 412678834162848948603, 44573440429472131194781, 5170931768652930067543199, 641240112753392800506551457, 84648865815216502596932335523 (list; graph; refs; listen; history; text; internal format)
 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 MAPLE a := n -> (-1)^n*add ((-2)^k*(n+k)!/((n-k)!*k!), k = 0..n): seq(a(n), n = 0..16); 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: A331114 A245265 A143639 * A331656 A003717 A201697 Adjacent sequences:  A143409 A143410 A143411 * A143413 A143414 A143415 KEYWORD easy,nonn AUTHOR Peter Bala, Aug 14 2008 STATUS approved

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Last modified February 26 03:29 EST 2020. Contains 332273 sequences. (Running on oeis4.)