A034323 a(n) = n-th quintic factorial number divided by 2.

1, 7, 84, 1428, 31416, 848232, 27143424, 1004306688, 42180880896, 1982501402112, 103090072909824, 5876134155859968, 364320317663318016, 24409461283442307072, 1757481212407846109184, 135326053355404150407168, 11096736375143140333387776, 965416064637453209004736512

Offset

1,2

Links

Robert Israel, Table of n, a(n) for n = 1..355

J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014-2020.

Formula

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

E.g.f.: (-1 + (1-5*x)^(-2/5))/2, with a(0) = 0.

a(n) ~ sqrt(2*Pi) * 5/(2*Gamma(2/5)) * n^(9/10) * (5*n/e)^n * (1 + (109/300)/n - ...). - Joe Keane (jgk(AT)jgk.org), Nov 24 2001

D-finite with recurrence a(n) = (5*n-3)*a(n-1). - Robert Israel, Feb 10 2019

From Amiram Eldar, Dec 19 2022: (Start)

a(n) = A047055(n)/2.

Sum_{n>=1} 1/a(n) = 2*(e/5^3)^(1/5)*(Gamma(2/5) - Gamma(2/5, 1/5)). (End)

Maple

f:= gfun:-rectoproc({a(n)=(5*n-3)*a(n-1), a(1)=1}, a(n), remember):
map(f, [$1..40]); # Robert Israel, Feb 10 2019

Mathematica

Table[Product[5j-3, {j, n}]/2, {n, 20}] (* Harvey P. Dale, Nov 25 2013 *)

Prog

(PARI) vector(20, n, prod(j=1, n, 5*j-3)/2) \\ G. C. Greubel, Feb 10 2019
(Magma) [(&*[5*j-3: j in [1..n]])/2: n in [1..20]]; // G. C. Greubel, Feb 10 2019
(Sage) [product(5*j-3 for j in (1..n))/2 for n in (1..20)] # G. C. Greubel, Feb 10 2019
(GAP) a:=[1];; for n in [2..20] do a[n]:=(5*n-3)*a[n-1]; od; a; # G. C. Greubel, Feb 10 2019

Crossrefs

Cf. A008548, A034300, A034301, A034325, A047055.

Sequence in context: A346582 A234510 A341966 * A172455 A258174 A370934

Adjacent sequences: A034320 A034321 A034322 * A034324 A034325 A034326

Keyword

easy,nonn

Author

Wolfdieter Lang

Status

approved