A055842 Expansion of (1-x)^2/(1-5*x).

1, 3, 16, 80, 400, 2000, 10000, 50000, 250000, 1250000, 6250000, 31250000, 156250000, 781250000, 3906250000, 19531250000, 97656250000, 488281250000, 2441406250000, 12207031250000, 61035156250000, 305175781250000, 1525878906250000, 7629394531250000

Offset: 0

Author

Barry E. Williams, May 30 2000

Keywords

• easy
• nonn

Comments

First differences of A005054.

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

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

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Milan Janjic, Enumerative Formulas for Some Functions on Finite Sets
• Index entries for linear recurrences with constant coefficients, signature (5).

Crossrefs

Cf. A000351, A005054.

Programs

GAP

Concatenation([1,3], List([2..30], n-> 16*5^(n-2) )); # G. C. Greubel, Jan 21 2020

Magma

[1,3] cat [16*5^(n-2): n in [2..30]]; // G. C. Greubel, Jan 21 2020

Magma

R:=PowerSeriesRing(Rationals(), 25); Coefficients(R!( (1-x)^2/(1-5*x))); // Marius A. Burtea, Jan 21 2020

Maple

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

Mathematica

Join[{1,3},16 5^(Range[2,30]-2)] (* Harvey P. Dale, Apr 03 2013 *)

PARI

Vec((1-x)^2/(1-5*x) + O(x^30)) \\ Altug Alkan, Mar 13 2016

Sage

[1,3]+[16*5^(n-2) for n in (2..30)] # G. C. Greubel, Jan 21 2020

Formula

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

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

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

a(n) = Sum_{k=0..n} A201780(n,k)*3^k. - Philippe Deléham, Dec 05 2011

E.g.f.: (9 - 5*x + 16*exp(x))/25. - G. C. Greubel, Jan 21 2020