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Quadruple factorial numbers n!!!!: a(n) = n*a(n-4).
(Formerly M0534)
63

%I M0534 #36 Sep 08 2022 08:44:35

%S 1,1,2,3,4,5,12,21,32,45,120,231,384,585,1680,3465,6144,9945,30240,

%T 65835,122880,208845,665280,1514205,2949120,5221125,17297280,40883535,

%U 82575360,151412625,518918400,1267389585,2642411520,4996616625

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

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.

%H Klaus Brockhaus, <a href="/A007662/b007662.txt">Table of n, a(n) for n = 0..500</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>.

%F 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

%F Sum_{n>=0} 1/a(n) = A288091. - _Amiram Eldar_, Nov 10 2020

%t 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_ *)

%t 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 *)

%o (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

%o (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

%o (PARI) a(n)=if(n<6,max(n,1),n*a(n-4)) \\ _Charles R Greathouse IV_, Jun 23 2011

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

%K nonn

%O 0,3

%A _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_