A007662 Quadruple factorial numbers n!!!!: a(n) = n*a(n-4). (Formerly M0534)

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

0,3

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.

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A367408 A117577 A109849 * A348370 A069469 A175807

Adjacent sequences: A007659 A007660 A007661 * A007663 A007664 A007665

Keyword

nonn

Author

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

Status

approved