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%I #16 Mar 26 2023 14:30:55
%S 1,1,-4,-3,-16,-15,144,105,1024,945,-14400,-10395,-147456,-135135,
%T 2822400,2027025,37748736,34459425,-914457600,-654729075,-15099494400,
%U -13749310575,442597478400,316234143225,8697308774400,7905853580625,-299195895398400,-213458046676875,-6818690079129600,-6190283353629375
%N Row 4 of array in A288580.
%D F. Smarandache, Back and Forth Factorials, Arizona State Univ., Special Collections, 1972.
%H J. Dezert, ed., <a href="http://www.mathematicsmagazine.com/corresp/J_Dezert/JDezert.htm">Smarandacheials (1)</a>, Mathematics Magazine for Grades 1-12, No. 4, 2004.
%H J. Dezert, ed., <a href="http://www.mathematicsmagazine.com/corresp/J_Dezert/JDezert2.htm">Smarandacheials (2)</a>, Mathematics Magazine for Grades 1-12, No. 4, 2004.
%F a(n) = !n!_4 = Prod_{i=0, 1, 2, ... .}_{0<|n-4i|<=n}(n-4i) = n*(n-4)*(n-8)....
%e !9!_4 = 9*(9-4)*(9-8)*(9-12)*(9-16) = 9*(5)*(1)*(-3)*(-7) = 945.
%p T:=proc(n,k) local i,p;
%p p:=1;
%p for i from 0 to floor(2*n/k) do
%p if n-k*i <> 0 then p:=p*(n-k*i) fi; od:
%p p;
%p end;
%p r:=k->[seq(T(n,k), n=0..60)]; r(4); # _N. J. A. Sloane_, Jul 03 2017
%t T[n_, k_] := Module[{i, p = 1},
%t For[i = 0, i <= Floor[2 n/k], i++,
%t If[n - k i != 0, p *= (n - k i)]]; p];
%t T[_, 0] = 1;
%t a[n_] := T[n, 4];
%t Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, Feb 16 2023 *)
%Y Cf. A288580.
%K sign
%O 0,3
%A J. Dezert (Jean.Dezert(AT)onera.fr), Mar 21 2004
%E Entry revised by _N. J. A. Sloane_, Jul 03 2017
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