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A052521 Number of pairs of sequences of cardinality at least 3.

%I A052521 #29 May 28 2025 20:02:14
%S A052521 0,0,0,0,0,0,720,10080,120960,1451520,18144000,239500800,3353011200,
%T A052521 49816166400,784604620800,13076743680000,230150688768000,
%U A052521 4268249137152000,83230858174464000,1703031405723648000
%N A052521 Number of pairs of sequences of cardinality at least 3.
%H A052521 G. C. Greubel, <a href="/A052521/b052521.txt">Table of n, a(n) for n = 0..445</a>
%H A052521 Milan Janjic, <a href="http://www.pmfbl.org/janjic/">Enumerative Formulas for Some Functions on Finite Sets</a>.
%H A052521 INRIA Algorithms Project, <a href="http://ecs.inria.fr/services/structure?nbr=88">Encyclopedia of Combinatorial Structures 88</a>.
%F A052521 E.g.f.: x^6/(1-x)^2.
%F A052521 (n-5)*a(n+1) + (4 + 3*n - n^2)*a(n) = 0, with a(0) = a(1) = a(2) = a(3) = a(4) = a(5) = 0, a(6) = 720.
%F A052521 a(n) = (n-5)*n!.
%F A052521 From _Amiram Eldar_, Jan 14 2021: (Start)
%F A052521 Sum_{n>=6} 1/a(n) = 5477/7200 - 17*e/60 - gamma/120 + Ei(1)/120 = 5477/7200 - (17/60)*A001113 - (1/120)*A001620 + A091725/120.
%F A052521 Sum_{n>=6} (-1)^n/a(n) = 403/7200 - 1/(6*e) + gamma/120 - Ei(-1)/120 = 403/7200 - (1/6)*A068985 + (1/120)*A001620 + (1/120)*A099285. (End)
%p A052521 spec := [S,{B=Sequence(Z,3 <= card), S=Prod(B,B)},labeled]: # Pairs spec
%p A052521 seq(combstruct[count](spec, size=n), n=0..20);
%t A052521 Table[If[n<6, 0, (n-5)*n!], {n,0,20}] (* _G. C. Greubel_, May 13 2019 *)
%o A052521 (PARI) {a(n) = if(n<6, 0, (n-5)*n!)}; \\ _G. C. Greubel_, May 13 2019
%o A052521 (Magma) [n le 5 select 0 else (n-5)*Factorial(n): n in [0..20]]; // _G. C. Greubel_, May 13 2019
%o A052521 (Sage) [0,0,0,0,0,0]+[(n-5)*factorial(n) for n in (6..20)] # _G. C. Greubel_, May 13 2019
%o A052521 (GAP) Concatenation([0,0,0,0,0,0], List([6..20], n-> (n-5)*Factorial(n))); # _G. C. Greubel_, May 13 2019
%Y A052521 Cf. sequences with formula (n + k)*n! listed in A282466.
%Y A052521 Cf. A001113, A001620, A068985, A091725, A099285.
%K A052521 nonn,easy
%O A052521 0,7
%A A052521 encyclopedia(AT)pommard.inria.fr, Jan 25 2000