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A142983 a(1) = 1, a(2) = 2, a(n+2) = 2*a(n+1) + (n+1)*(n+2)*a(n). 13
1, 2, 10, 44, 288, 1896, 15888, 137952, 1419840, 15255360, 186693120, 2387093760, 33898314240, 502247692800, 8123141376000, 136785729024000, 2483065912320000, 46822564905984000, 942853671825408000, 19678282007924736000, 435355106182520832000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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), normalized 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 polynomial.

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 behavior 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.

REFERENCES

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Hyun Kwang Kim, On Regular Polytope Numbers, Proc. Amer. Math. Soc., 131 (2002), 65-75.

Weisstein Eric, W. Meixner polynomial of the first kind

Weisstein Eric, W. Mittag-Leffler polynomial

FORMULA

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 behavior 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;

E.g.f.: (2*log(x+1)-x)/(x-1)^2. - Vaclav Kotesovec, Oct 21 2012

MAPLE

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

MATHEMATICA

Rest[CoefficientList[Series[(-x+2*Log[x+1])/(x-1)^2, {x, 0, 20}], x]*Range[0, 20]!] (* Vaclav Kotesovec, Oct 21 2012 *)

PROG

(Haskell)

a142983 n = a142983_list !! (n-1)

a142983_list = 1 : 2 : zipWith (+)

                       (map (* 2) $ tail a142983_list)

                       (zipWith (*) (drop 2 a002378_list) a142983_list)

-- Reinhard Zumkeller, Jul 17 2015

CROSSREFS

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

Cf. A002378.

Sequence in context: A105485 A151313 A144896 * A227852 A065805 A145239

Adjacent sequences:  A142980 A142981 A142982 * A142984 A142985 A142986

KEYWORD

easy,nonn

AUTHOR

Peter Bala, Jul 17 2008

STATUS

approved

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Last modified May 17 06:30 EDT 2021. Contains 343965 sequences. (Running on oeis4.)