A049016 Expansion of 1/((1-x)^5 - x^5).

1, 5, 15, 35, 70, 127, 220, 385, 715, 1430, 3004, 6385, 13380, 27370, 54740, 107883, 211585, 416405, 826045, 1652090, 3321891, 6690150, 13455325, 26985675, 53971350, 107746282, 214978335, 429124630, 857417220, 1714834440, 3431847189

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn
• easy

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000
• Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,2).

Crossrefs

Sequences of the form 1/((1-x)^m - x^m): A000079 (m=1,2), A024495 (m=3), A000749 (m=4), this sequence (m=5), A192080 (m=6), A049017 (m=7), A290995 (m=8), A306939 (m=9).

Cf. A000750, A078789.

Programs

Magma

R:=PowerSeriesRing(Integers(), 30); Coefficients(R!( 1/((1-x)^5-x^5) )); // G. C. Greubel, Apr 11 2023

Mathematica

CoefficientList[Series[1/((1-x)^5-x^5),{x,0,30}],x] (* or *) LinearRecurrence[ {5,-10,10,-5,2},{1,5,15,35,70},40] (* Harvey P. Dale, Jan 20 2014 *)

SageMath

def A049016_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( 1/((1-x)^5-x^5) ).list()
A049016_list(30) # G. C. Greubel, Apr 11 2023

Formula

G.f.: 1/((1-x)^5-x^5) = 1/( (1-2*x)*(1-3*x+4*x^2-2*x^3+x^4) ).

a(10*n+3) = A078789(5*n+3).

a(10*n+5) = A078789(5*n+4).

a(n) = (-1)^n * A000750(n).

Binomial transform of expansion of (1+x)^4/(1-x^5), or (1, 4, 6, 4, 1, 1, 4, 6, 4, 1, ...). - Paul Barry, Mar 19 2004

a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + 2*a(n-5). - Paul Curtz, May 24 2008

G.f.: -1/( x^5 - 1 + 5*x/Q(0) ) where Q(k) = 1 + k*(x+1) + 5*x - x*(k+1)*(k+6)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 15 2013