A024217 a(n) = ( Product {k = 1..n} 3*k - 2 ) * ( Sum {k = 1..n} (-1)^(k+1)/(3*k - 2) ).

1, 3, 25, 222, 3166, 47016, 951544, 19827408, 520029520, 13952218560, 449559799360, 14756761434240, 563961412362880, 21893890640563200, 968019931702297600, 43385863589508249600, 2178487766250586470400, 110704921777161066700800, 6222745685273069016064000

Offset

1,2

Comments

Original name was: s(1)*s(2)*...*s(n)(1/s(1) - 1/s(2) + ... + c/s(n)), where c=(-1)^(n+1) and s(k) = 3k-2 for k = 1,2,3,...

Links

Table of n, a(n) for n=1..19.

L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 5

Formula

From Peter Bala, Feb 20 2015: (Start)

a(n) = A007559(n) * Sum {k = 1..n} (-1)^(k+1)/(3*k - 2).

Recurrence: a(n+1) = 3*a(n) + (3*n - 2)^2*a(n-1) with a(1) = 1 and a(2) = 3.

The triple factorial numbers A007559 also satisfy this second-order recurrence equation. This leads to the continued fraction representation a(n)/A007559(n) = 1/(1 + 1^2/(3 + 4^2/(3 + 7^2/(3 + ... + (3*n - 2)^2/(3 ))))).

Taking the limit as n -> infinity gives the generalized continued fraction: Sum {k >= 1} (-1)^(k+1)/(3*k - 2) = 1/(1 + 1^2/(3 + 4^2/(3 + 7^2/(3 + 10^2/(3 + ... ))))) due to Euler.

The alternating sum has the value 1/3*( log(2) + Pi/sqrt(3) ) = A113476. Cf. A024396. (End)

a(n) ~ sqrt(2*Pi) * (sqrt(3)*Pi + 3*log(2)) * 3^(n-2) * n^(n-1/6) / (GAMMA(1/3) * exp(n)). - Vaclav Kotesovec, Feb 21 2015

Maple

a[1] := 1: a[2] := 3: for n from 3 to 20 do a[n] := 3*a[n-1]+(3*n-5)^2*a[n-2] end do: seq(a[n], n = 1 .. 20); # Peter Bala, Feb 20 2015

Mathematica

Table[Product[3*k-2, {k, 1, n}] * Sum[(-1)^(k+1)/(3*k-2), {k, 1, n}], {n, 1, 20}] (* Vaclav Kotesovec, Feb 21 2015 *)

Prog

(Magma) I:=[1, 3]; [n le 2 select I[n] else 3*Self(n-1)+(3*n-5)^2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 21 2015

Crossrefs

Cf. A007559, A024199, A024383, A024396, A113476.

Sequence in context: A112240 A155640 A370104 * A199679 A118726 A066221

Adjacent sequences: A024214 A024215 A024216 * A024218 A024219 A024220

Keyword

nonn,easy

Author

Clark Kimberling

Extensions

New name from Peter Bala, Feb 20 2015

a(18)-a(19) from Vincenzo Librandi, Feb 21 2015

Status

approved