1,2

This is the case m = 1 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) (we suppress the dependence of a(n) on m), which arises when accelerating the convergence of the series 1/2 + 1/2*sum {k = 1..inf} (-1)^(k+1)/(k*(k+1)) = log(2). For other cases see A142984 (m=2), A142985 (m=3), A142986 (m=4) and A142987 (m=5). The solution to the general recurrence may be expressed as a sum: a(n) = n!*p_m(n+1)*sum {k = 1..n} (-1)^(k+1)/(p_m(k)*p_m(k+1)), where p_m(x) = sum {k = 1..m} 2^(k-1)*C(m-1,k-1)*C(x,k) is the polynomial that gives the regular polytope numbers for the m-dimensional cross polytope as defined by [Kim](see A142978). The first few values are p_1(x) = x, p_2(x) = x^2, p_3(x) = (2*x^3+x)/3 and p_4(x) = (x^4+2*x^2)/3.

The polynomial p_m(x) is the unique polynomial solution of the difference equation x*(f(x+1)-f(x-1)) = 2*m*f(x), normalised so that f(1) = 1. The o.g.f. for the p_m(x) is 1/2*((1+t)/(1-t))^x = 1/2 + x*t + x^2*t^2 + (2*x^3+x)/3*t^3 + ... . Thus p_m(x) is, apart from a constant factor, the Meixner polynomial of the first kind M_m(x;b,c) at b = 0, c = -1, also known as a Mittag-Leffler poynomial.

The general recurrence in the first paragraph above has a second solution b(n) = n!*p_m(n+1) with b(1) = 2*m, b(2) = m^2+2. Hence the behaviour of a(n) for large n is given by lim n-> infinity a(n)/b(n) = sum {k = 1..inf} (-1)^(k+1)/(p_m(k)*p_m(k+1)) = 1/((2*m)+ 1*2/((2*m)+ 2*3/((2*m)+ 3*4/((2*m)+...+ *n*(n+1)/((2*m)+...))))) = 1 + (-1)^(m+1) * (2*m)*(log(2) - (1-1/2+1/3- ...+ (-1)^(m+1)/m)), where the final equality follows by a result of Ramanujan (see [Berndt, Chapter 12, Entry 32(i)]).

See A142979, A142988 and A142992 for similar results. For corresponding results for Napier's constant e, the constant zeta(2) and Apery's constant zeta(3) refer to A000522, A142995 and A143003 respectively.

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

Hyun Kwang Kim, On regular polytope numbers

Weisstein Eric, W. Meixner polynomial of the first kind

Weisstein Eric, W. Mittag-Leffler polynomial

a(n) = n!*p(n+1)*sum {k = 1..n} (-1)^(k+1)/(p(k)*p(k+1)), where p(n) = n. Recurrence: a(1) = 1, a(2) = 2, a(n+2) = 2*a(n+1)+(n+1)*(n+2)*a(n). The sequence b(n):= n!*p(n+1) satisfies the same recurrence with b(1) = 2, b(2) = 6. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(2 +1*2/(2 +2*3/(2 +3*4/(2 +...+(n-1)*n/2)))), for n >=2. The behaviour of a(n) for large n is given by lim n -> infinity a(n)/b(n) = 1/(2 +1*2/(2 +2*3/(2 +3*4/(2 +...+n*(n+1)/(2 +...))))) = sum {k = 1..inf} (-1)^(k+1)/(k*(k+1)) = 2*log(2) - 1;

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

Cf. A000522, A142979, A142984, A142985, A142986, A142987, A142988, A142992.

Sequence in context: A105485 A151313 A144896 this_sequence A065805 A145239 A068561

Adjacent sequences: A142980 A142981 A142982 this_sequence A142984 A142985 A142986

easy,nonn

Peter Bala (pbala(AT)toucansurf.com), Jul 17 2008