A034300 a(n) = n-th quintic factorial number divided by 3.

1, 8, 104, 1872, 43056, 1205568, 39783744, 1511782272, 65006637696, 3120318609408, 165376886298624, 9591859405320192, 604287142535172096, 41091525692391702528, 2999681375544594284544, 233975147292478354194432, 19419937225275703398137856, 1708954475824261899036131328

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..350

Formula

3*a(n) = (5*n-2)(!^5) = Product_{j=1..n} (5*j-2) = A047056(n).

E.g.f.: (1-5*x)^(-3/5)/3.

a(n) ~ sqrt(2*Pi) * 5/(3*Gamma(3/5)) * n^(11/10) * (5*n/e)^n * (1 + (169/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

a(n) = 5^n * Pochhammer(3/5, n)/3. - G. C. Greubel, Aug 23 2019

D-finite with recurrence: a(n) +(-5*n+2)*a(n-1)=0. - R. J. Mathar, Feb 20 2020

Sum_{n>=1} 1/a(n) = 3*(e/5^2)^(1/5)*(Gamma(3/5) - Gamma(3/5, 1/5)). - Amiram Eldar, Dec 19 2022

Maple

a:= n-> mul(5*k-2, k=1..n)/3; seq(a(n), n=1..25); # G. C. Greubel, Aug 23 2019

Mathematica

Table[Product[5j-2, {j, n}], {n, 20}]*1/3 (* Harvey P. Dale, Jul 25 2011 *)

Prog

(PARI) a(n) = prod(k=1, n, 5*k-2)/3;
vector(25, n, a(n)) \\ G. C. Greubel, Aug 23 2019
(Magma) [(&*[5*k-2: k in [1..n]])/3: n in [1..25]]; // G. C. Greubel, Aug 23 2019
(Sage) [5^n*rising_factorial(3/5, n)/3 for n in (1..25)] # G. C. Greubel, Aug 23 2019
(GAP) List([1..25], n-> Product([1..n], k-> 5*k-2)/3 ); # G. C. Greubel, Aug 23 2019

Crossrefs

Cf. A008548, A034301, A047056.

Sequence in context: A106260 A112121 A141383 * A266044 A359801 A266040

Adjacent sequences: A034297 A034298 A034299 * A034301 A034302 A034303

Keyword

nonn

Author

Wolfdieter Lang

Extensions

Terms a(17) onward added by G. C. Greubel, Aug 23 2019

Status

approved