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 A024396 a(n) = ( Product {k = 1..n} 3*k - 1 ) * ( Sum {k = 1..n} (-1)^(k+1)/(3*k - 1) ). 4
 1, 3, 34, 294, 4996, 72612, 1661680, 34029840, 981118240, 25947526560, 902963019520, 29279156256000, 1193967167680000, 45861003136704000, 2144641818280192000, 95220827527499520000, 5023176259163442688000, 253121597596239128064000, 14869466904778827894784000 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Original name: s(1)*s(2)*...*s(n)*(1/s(1) - 1/s(2) + ... + c/s(n)), where c = (-1)^(n+1) and s(k) = 3k-1 for k = 1,2,3,... LINKS L. Euler, De fractionibus continuis observationes, The Euler Archive, Index Number 123, Section 5 FORMULA From Peter Bala, Feb 20 2015: (Start) Recurrence: a(n+1) = 3*a(n) + (3*n - 1)^2*a(n-1) with a(1) = 1 and a(2) = 3. The triple factorial numbers A008544 satisfy the same second-order recurrence equation. This leads to the continued fraction representation a(n)/A008544(n) = 1/(2 + 2^2/(3 + 5^2/(3 + 8^2/(3 + ... + (3*n - 1)^2/(3 )))). Taking the limit as n -> infinity gives the generalized continued fraction: Sum {k >= 1} (-1)^(k+1)/(3*k - 1) = 1/(2 + 2^2/(3 + 5^2/(3 + 8^2/(3 + 11^2/(3 + ... ))))) due to Euler. The alternating sum has the value 1/3*( Pi/sqrt(3) - log(2) ) = A193534. Cf. A024217. (End) a(n) ~ GAMMA(1/3) * 3^(n-1) * n^(n+1/6) * (Pi - sqrt(3)*log(2)) / (sqrt(2*Pi) * 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-4)^2*a[n-2] end do: seq(a[n], n = 1 .. 20); # Peter Bala, Feb 20 2015 MATHEMATICA Table[Product[3*k-1, {k, 1, n}] * Sum[(-1)^(k+1)/(3*k-1), {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-4)^2*Self(n-2): n in [1..20]]; // Vincenzo Librandi, Feb 21 2015 CROSSREFS Cf. A008544, A024217, A193534. Sequence in context: A266696 A141789 A121077 * A246384 A066265 A268802 Adjacent sequences:  A024393 A024394 A024395 * A024397 A024398 A024399 KEYWORD nonn,easy AUTHOR EXTENSIONS New name from Peter Bala, Feb 20 2015 a(18)-a(19) from Vincenzo Librandi, Feb 21 2015 STATUS approved

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Last modified July 25 19:05 EDT 2021. Contains 346291 sequences. (Running on oeis4.)