A055842 - OEIS

%I #34 Sep 08 2022 08:45:01

%S 1,3,16,80,400,2000,10000,50000,250000,1250000,6250000,31250000,

%T 156250000,781250000,3906250000,19531250000,97656250000,488281250000,

%U 2441406250000,12207031250000,61035156250000,305175781250000,1525878906250000,7629394531250000

%N Expansion of (1-x)^2/(1-5*x).

%C First differences of A005054.

%C For n>=2, a(n) is equal to the number of functions f:{1,2,...,n}->{1,2,3,4,5} such that for fixed, different x_1, x_2 in {1,2,...,n} and fixed y_1, y_2 in {1,2,3,4,5} we have f(x_1)<>y_1 and f(x_2)<> y_2. - _Milan Janjic_, Apr 19 2007

%C a(n) is the number of generalized compositions of n when there are 4 *i-1 different types of i, (i=1,2,...). - _Milan Janjic_, Aug 26 2010

%D A. H. Beiler, Recreations in the Theory of Numbers, Dover, N.Y., 1964, pp. 194-196.

%H G. C. Greubel, <a href="/A055842/b055842.txt">Table of n, a(n) for n = 0..1000</a>

%H Milan Janjic, <a href="https://pmf.unibl.org/wp-content/uploads/2017/10/enumfun.pdf">Enumerative Formulas for Some Functions on Finite Sets</a>

%H <a href="/index/Rec#order_01">Index entries for linear recurrences with constant coefficients</a>, signature (5).

%F a(n) = 16*5^(n-2), a(0)=1, a(1)=3.

%F a(n) = 5*a(n-1) + (-1)^n*binomial(2,2-n).

%F G.f.: (1-x)^2/(1-5*x).

%F a(n) = Sum_{k=0..n} A201780(n,k)*3^k. - _Philippe Deléham_, Dec 05 2011

%F E.g.f.: (9 - 5*x + 16*exp(x))/25. - _G. C. Greubel_, Jan 21 2020

%p seq( `if`(n<2, 2*n+1, 16*5^(n-2)), n=0..30); # _G. C. Greubel_, Jan 21 2020

%t Join[{1,3},16 5^(Range[2,30]-2)] (* _Harvey P. Dale_, Apr 03 2013 *)

%o (PARI) Vec((1-x)^2/(1-5*x) + O(x^30)) \\ _Altug Alkan_, Mar 13 2016

%o (Magma) [1,3] cat [16*5^(n-2): n in [2..30]]; // _G. C. Greubel_, Jan 21 2020

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-x)^2/(1-5*x))); // _Marius A. Burtea_, Jan 21 2020

%o (Sage) [1,3]+[16*5^(n-2) for n in (2..30)] # _G. C. Greubel_, Jan 21 2020

%o (GAP) Concatenation([1,3], List([2..30], n-> 16*5^(n-2) )); # _G. C. Greubel_, Jan 21 2020

%Y Cf. A000351, A005054.

%K easy,nonn

%O 0,2

%A _Barry E. Williams_, May 30 2000