A142984 a(1) = 1, a(2) = 4, a(n+2) = 4*a(n+1) + (n + 1)*(n + 2)*a(n).

1, 4, 22, 136, 984, 8016, 73392, 742464, 8254080, 99838080, 1307301120, 18407831040, 277570298880, 4460506444800, 76131788544000, 1375048700928000, 26208041287680000, 525597067634688000, 11065538390925312000

Offset

1,2

Comments

This is the case m = 2 of the general recurrence a(1) = 1, a(2) = 2*m, a(n+2) = 2*m*a(n+1) + (n + 1)*(n + 2)*a(n), which arises when accelerating the convergence of a certain series for the constant log(2). See A142983 for remarks on the general case.

References

Bruce C. Berndt, Ramanujan's Notebooks Part II, Springer-Verlag.

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..250

Formula

a(n) = n!*p(n+1)*Sum_{k = 1..n} (-1)^(k+1)/(p(k)*p(k+1)), where p(n) = n^2.

Recurrence: a(1) = 1, a(2) = 4, a(n+2) = 4*a(n+1) + (n + 1)*(n + 2)*a(n).

The sequence b(n) := n!*p(n+1) satisfies the same recurrence with b(1) = 4 and b(2) = 18.

Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(4 + 1*2/(4 + 2*3/(4 + 3*4/(4 + ... + n*(n - 1)/(4))))), for n >= 2.

The behavior of a(n) for large n is given by lim_{n -> oo} a(n)/b(n) = Sum_{k >= 1} (-1)^(k+1)/(k^2*(k + 1)^2) = 1/(4 + 1*2/(4 + 2*3/(4 + 3*4/(4 + ... + n*(n - 1)/(4 + ...))))) = 3 - 4*log(2), where the final equality follows by a result of Ramanujan (see [Berndt, Chapter 12, Entry 32(i)]).

a(n) = n! * (((-1)^n * (2 * Psi(n/2 + 1) - 2 * Psi(n/2 + 3/2)) - 4 * log(2) + 3) * (n+1)^2 + (-1)^n * (2 * n + 1)). - Robert Israel, Mar 07 2024

Maple

a := n -> n!*(n+1)^2*sum ((-1)^(k+1)/(k^2*(k+1)^2), k = 1..n): seq(a(n), n = 1..20);

Prog

(Haskell)
a142984 n = a142984_list !! (n-1)
a142984_list = 1 : 4 : zipWith (+)
                       (map (* 4) $ tail a142984_list)
                       (zipWith (*) (drop 2 a002378_list) a142984_list)
-- Reinhard Zumkeller, Jul 17 2015

Crossrefs

Cf. A142983, A142985, A142986, A142987.

Cf. A002378.

Sequence in context: A069835 A007196 A091638 * A283055 A097593 A188686

Adjacent sequences: A142981 A142982 A142983 * A142985 A142986 A142987

Keyword

easy,nonn

Author

Peter Bala, Jul 17 2008

Status

approved