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

%I A007662 M0534 #37 Feb 16 2025 08:32:31
%S A007662 1,1,2,3,4,5,12,21,32,45,120,231,384,585,1680,3465,6144,9945,30240,
%T A007662 65835,122880,208845,665280,1514205,2949120,5221125,17297280,40883535,
%U A007662 82575360,151412625,518918400,1267389585,2642411520,4996616625
%N A007662 Quadruple factorial numbers n!!!!: a(n) = n*a(n-4).
%D A007662 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007662 J. Spanier and K. B. Oldham, An Atlas of Functions, Hemisphere, NY, 1987, p. 23.
%H A007662 Klaus Brockhaus, <a href="/A007662/b007662.txt">Table of n, a(n) for n = 0..500</a>
%H A007662 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Multifactorial.html">Multifactorial</a>.
%F A007662 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 A007662 Sum_{n>=0} 1/a(n) = A288091. - _Amiram Eldar_, Nov 10 2020
%t A007662 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 A007662 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 A007662 (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 A007662 (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 A007662 (PARI) a(n)=if(n<6,max(n,1),n*a(n-4)) \\ _Charles R Greathouse IV_, Jun 23 2011
%Y A007662 Cf. A047053, A007696, A001813, A008545, A000142, A006882, A007661, A288091.
%K A007662 nonn
%O A007662 0,3
%A A007662 _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_