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a(n) is the number of n-digit positive integers with exactly four distinct base 10 digits.
0

%I #13 Oct 19 2020 11:10:10

%S 0,0,0,4536,45360,294840,1587600,7715736,35244720,154700280,661122000,

%T 2773768536,11487556080,47136955320,192126589200,779279814936,

%U 3149513947440,12695388483960,51073849285200,205172877726936,823325141746800,3301203837670200,13228529919066000

%N a(n) is the number of n-digit positive integers with exactly four distinct base 10 digits.

%C a(n) is the number of n-digit numbers in A031969.

%H <a href="/index/Rec#order_04">Index entries for linear recurrences with constant coefficients</a>, signature (10,-35,50,-24).

%H <a href="/index/Di#digits">Index entries for sequences related to digits</a>.

%F O.g.f.: 4536*x^4/(1 - 10*x + 35*x^2 - 50*x^3 + 24*x^4).

%F E.g.f.: 189*(exp(x) - 1)^4.

%F a(n) = 10*a(n-1) - 35*a(n-2) + 50*a(n-3) - 24*a(n-4) for n > 4.

%F a(n) = 4536*S2(n, 4) where S2(n, 4) = A000453(n).

%F a(n) = 189*(4^n - 4*3^n + 3*2^(n+1) - 4).

%F a(n) ~ 189 * 4^n.

%F a(n) = 189*(A000302(n) - 4*A000244(n) + 3*A000079(n+1) - 4).

%F a(n) = A337127(n, 4).

%e a(1) = a(2) = a(3) = 0 since the positive integers must have at least four digits;

%e a(4) = #{wxyz in N | w,x,y,z are four different digits with w != 0} = A073531(4) = 4536;

%e a(5) = 45360 since #[99999] - #[9999] - #(11111*[9]) - A335843(5) - A337313(5) - #{vwxyz in N | v,w,x,y,z are five different digits with v != 0} = 99999 - 9999 - 9 - 1215 - 16200 - 9*9*8*7*6 = 45360;

%e ...

%t LinearRecurrence[{10,-35,50,-24},{0,0,0,4536},23]

%o (PARI) concat([0,0,0],Vec(4536*x^4/(1-10*x+35*x^2-50*x^3+24*x^4)+O(x^24)))

%Y Cf. A000079, A000244, A000302, A000453, A031969, A073531, A335843, A337313, A337127.

%Y Cf. A055642, A180599, A235155, A235691, A235718.

%K nonn,easy,base

%O 1,4

%A _Stefano Spezia_, Sep 26 2020