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Number of n-digit decimal numbers with leading zeros allowed where no digit appears exactly twice.
3

%I #41 Oct 07 2024 10:41:25

%S 10,90,730,5410,37900,264250,1908910,14322520,110305720,875799550,

%T 7203731050,60866700940,527138423380,4696469283970,42797376850150,

%U 397903278667600,3783002076771760,36604891556846470,360117919584474850,3603555076712108500

%N Number of n-digit decimal numbers with leading zeros allowed where no digit appears exactly twice.

%H Math Stackexchange, Marko Riedel et al., <a href="http://math.stackexchange.com/questions/1765530/">N digit numbers with no digit appearing exactly twice.</a>

%F E.g.f.: Sum_{q=1..N} C(10,q)*(exp(z)-z^2/2-1)^q.

%F E.g.f.: (exp(z)-z^2/2)^10.

%e Computing the value for n=1 the ten one-digit numbers 0-9 all qualify.

%e Computing the value for n=2 of two-digit numbers 00-99 all qualify except those consisting of a double digit (00, 11, ..., 99).

%p STNE2 := (n,k)->n!*coeftayl((exp(z)-z^2/2-1)^k/k!, z=0, n);

%p a(N) := N -> add(combinat:-binomial(10, q)*STNE2(N, q)*q!, q = 1 .. N)

%t With[{nn=20},Rest[CoefficientList[Series[(Exp[x]-x^2/2)^10,{x,0,nn}],x] Range[0,nn]!]] (* _Harvey P. Dale_, Oct 07 2024 *)

%Y Cf. A272501, A272502, A272503.

%K nonn,base

%O 1,1

%A _Marko Riedel_, Apr 30 2016