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A052520 Number of pairs of sequences of cardinality at least 2.

%I A052520 #41 May 28 2025 20:05:41
%S A052520 0,0,0,0,24,240,2160,20160,201600,2177280,25401600,319334400,
%T A052520 4311014400,62270208000,958961203200,15692092416000,271996268544000,
%U A052520 4979623993344000,96035605585920000,1946321606541312000,41359334139002880000,919636959090769920000,21356013827774545920000
%N A052520 Number of pairs of sequences of cardinality at least 2.
%H A052520 Vincenzo Librandi, <a href="/A052520/b052520.txt">Table of n, a(n) for n = 0..400</a>
%H A052520 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A052520 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=87">Encyclopedia of Combinatorial Structures 87</a>.
%F A052520 E.g.f.: x^4/(1-x)^2.
%F A052520 (n-3)*a(n+1) + (2+n-n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = 0, a(4) = 24.
%F A052520 a(n) = (n-3)*n!, n>2.
%F A052520 a(n) = (n+1)!*(n-3)/(n+1), n>2. - _Gary Detlefs_, Oct 02 2011
%F A052520 From _Amiram Eldar_, Jan 14 2021: (Start)
%F A052520 Sum_{n>=4} 1/a(n) = 59/36 - 2*e/3 - gamma/6 + Ei(1)/6 = 59/36 - (2/3)*A001113 - (1/6)*A001620 + A091725/2.
%F A052520 Sum_{n>=4} (-1)^n/a(n) = 1/36 - 1/(3*e) + gamma/6 - Ei(-1)/6 = 1/36 - (1/3)*A068985 + (1/6)*A001620 + (1/6)*A099285. (End)
%p A052520 spec := [S,{B=Sequence(Z,2 <= card),S=Prod(B,B)},labeled]: seq(combstruct[count](spec,size=n), n=0..20);
%t A052520 Table[Sum[n!, {i,4,n}], {n, 0, 19}] (* _Zerinvary Lajos_, Jul 12 2009 *)
%t A052520 With[{nn=20},CoefficientList[Series[x^4/(x-1)^2,{x,0,nn}],x] Range[0,nn]!] (* _Harvey P. Dale_, Jun 03 2016 *)
%o A052520 (PARI) {a(n) = if(n<4, 0, (n-3)*n!)}; \\ _G. C. Greubel_, May 13 2019
%o A052520 (Magma) [n le 3 select 0 else (n-3)*Factorial(n): n in [0..20]]; // _G. C. Greubel_, May 13 2019
%o A052520 (Sage) [0,0,0,0]+[(n-3)*factorial(n) for n in (4..20)] # _G. C. Greubel_, May 13 2019
%o A052520 (GAP) Concatenation([0,0,0,0], List([4..20], n-> (n-3)*Factorial(n))); # _G. C. Greubel_, May 13 2019
%Y A052520 Cf. sequences with formula (n + k)*n! listed in A282466.
%Y A052520 Cf. A001113, A001620, A068985, A091725, A099285.
%K A052520 easy,nonn
%O A052520 0,5
%A A052520 encyclopedia(AT)pommard.inria.fr, Jan 25 2000