A007662 Quadruple factorial numbers n!!!!: a(n) = n*a(n-4).

1, 1, 2, 3, 4, 5, 12, 21, 32, 45, 120, 231, 384, 585, 1680, 3465, 6144, 9945, 30240, 65835, 122880, 208845, 665280, 1514205, 2949120, 5221125, 17297280, 40883535, 82575360, 151412625, 518918400, 1267389585, 2642411520, 4996616625

Offset: 0

Author

N. J. A. Sloane, Robert G. Wilson v, Mira Bernstein

Keywords

• nonn

References

• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
• J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.

Links

• Klaus Brockhaus, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Multifactorial.

Crossrefs

Cf. A047053, A007696, A001813, A008545, A000142, A006882, A007661, A288091.

Programs

Magma

I:=[ 1, 1, 2, 3 ]; [ n le 4 select I[n] else (n-1)*Self(n-4): n in [1..36] ]; // Klaus Brockhaus, Jun 23 2011

Magma

A007662:=func< n | n eq 0 select 1 else &*[ k: k in [1..n] | k mod 4 eq n mod 4 ] >; [ A007662(n): n in [0..35] ]; // Klaus Brockhaus, Jun 23 2011

Mathematica

NFactorialM[n_Integer, m_Integer] := Block[{k = n, p = Max[1, n]}, While[k > m, k -= m; p *= k]; p]; Table[ NFactorialM[n, 4], {n, 0, 34}] (* Robert G. Wilson v *)
With[{k = 4}, Table[With[{q = Quotient[n + k - 1, k]}, k^q q! Binomial[n/k, q]], {n, 0, 34}]] (* Jan Mangaldan, Mar 21 2013 *)

PARI

a(n)=if(n<6,max(n,1),n*a(n-4)) \\ Charles R Greathouse IV, Jun 23 2011

Formula

a(n) ~ c * n^(n/4+1/2)/exp(n/4), where c = sqrt(Pi/2) if n=4*k, c = 2*sqrt(Pi)/Gamma(1/4) if n=4*k+1, c = sqrt(2) if n=4*k+2, c = sqrt(Pi)/Gamma(3/4) if n=4*k+3. - Vaclav Kotesovec, Jul 29 2013

Sum_{n>=0} 1/a(n) = A288091. - Amiram Eldar, Nov 10 2020