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A142986 a(1) = 1, a(2) = 8, a(n+2) = 8*a(n+1) + (n+1)*(n+2)*a(n). 5
1, 8, 70, 656, 6648, 72864, 862128, 10977408, 149892480, 2187106560, 33985025280, 560578268160, 9786290088960, 180315565516800, 3497645442816000, 71256899266560000, 1521414754578432000, 33975929212194816000 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

This is the case m = 4 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^4+2*n^2)/3 = A014820(n). Recurrence: a(1) = 1, a(2) = 8, a(n+2) = 8*a(n+1)+(n+1)*(n+2)*a(n). The sequence b(n):= n!*p(n+1) satisfies the same recurrence with b(1) = 8, b(2) = 66. Hence we obtain the finite continued fraction expansion a(n)/b(n) = 1/(8 +1*2/(8 +2*3/(8 +3*4/(8 +...+(n-1)*n/8)))), for n >=2. 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(k)*p(k+1)) = 1/(8 +1*2/(8 +2*3/(8 +3*4/(8 +...+n*(n+1)/(8 +...))))) = 17/3 - 8*log(2), where the final equality follows by a result of Ramanujan (see [Berndt, Chapter 12, Entry 32(i)]).

MAPLE

p := n -> (n^4+2*n^2)/3: a := n -> n!*p(n+1)*sum ((-1)^(k+1)/(p(k)*p(k+1)), k = 1..n): seq(a(n), n = 1..20);

MATHEMATICA

RecurrenceTable[{a[1]==1, a[2]==8, a[n]==8a[n-1]+n(n-1)a[n-2]}, a, {n, 20}] (* Harvey P. Dale, Apr 08 2015 *)

PROG

(Haskell)

a142986 n = a142986_list !! (n-1)

a142986_list = 1 : 8 : zipWith (+)

                       (map (* 8) $ tail a142986_list)

                       (zipWith (*) (drop 2 a002378_list) a142986_list)

-- Reinhard Zumkeller, Jul 17 2015

CROSSREFS

Cf. A014820, A142983, A142984, A142985, A142987.

Cf. A002378.

Sequence in context: A299175 A299938 A152263 * A123511 A322416 A287482

Adjacent sequences:  A142983 A142984 A142985 * A142987 A142988 A142989

KEYWORD

easy,nonn

AUTHOR

Peter Bala, Jul 17 2008

STATUS

approved

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Last modified May 18 17:28 EDT 2021. Contains 343996 sequences. (Running on oeis4.)