login

Reminder: The OEIS is hiring a new managing editor, and the application deadline is January 26.

Expansion of (1+x)^50 * (1-x)/(1 - x^51).
1

%I #17 Oct 02 2023 11:46:46

%S 1,49,1175,18375,210700,1888460,13771940,83993700,436994250,

%T 1968555050,7766844470,27081460630,84045912300,233460867500,

%U 582985137700,1312983918820,2672860120455,4923689695575,8206149492625,12352414499425

%N Expansion of (1+x)^50 * (1-x)/(1 - x^51).

%C From _G. C. Greubel_, Feb 16 2021: (Start)

%C Let a(n) be the coefficients of the expansion then a(n+51) = a(n) (i.e. periodic length 50) and a(m+26) = - a(25-m) for 0 <= m <= 24.

%C Expansions of the form (1+x)^m * (1-x)/(1 - x^(m+1)) have the coefficients a(n) = Sum_{j=0..(m+1)*n} ( binomial(m, n-(m+1)*j) - binomial(m, n-(m+1)*j-1) ). (End)

%H G. C. Greubel, <a href="/A173246/b173246.txt">Table of n, a(n) for n = 0..509</a>

%H <a href="/index/Rec#order_50">Index entries for linear recurrences with constant coefficients</a>, signature (-1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1).

%F G.f.: (1+x)^50 / Sum_{j=0..50} x^j.

%F From _G. C. Greubel_, Feb 16 2021: (Start)

%F G.f.: (1+x)^50 * (1-x)/(1 - x^51).

%F a(n) = (-1)*Sum_{j=1..50} a(n-j) for n > 50.

%F a(n) = Sum_{j=0..51*n} ( binomial(50, n-51*j) - binomial(50, n-51*j-1) ), n > 0. (End)

%p m:= 40;

%p S:= series( (1+x)^50*(1-x)/(1-x^51), x, m+1);

%p seq(coeff(S, x, j), j = 0..m); # _G. C. Greubel_, Feb 16 2021

%t CoefficientList[Series[(1+x)^50*(1-x)/(1-x^51), {x,0,40}], x] (* modified by _G. C. Greubel_, Feb 16 2021 *)

%o (Magma)

%o R<x>:=PowerSeriesRing(Integers(), 40);

%o Coefficients(R!( (1+x)^50*(1-x)/(1-x^51) )); // _G. C. Greubel_, Feb 16 2021

%o (Sage)

%o def A173246_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( (1+x)^50*(1-x)/(1-x^51) ).list()

%o A173246_list(40) # _G. C. Greubel_, Feb 16 2021

%Y Cf. A173245.

%K sign,easy,less

%O 0,2

%A _Roger L. Bagula_, Feb 13 2010

%E Edited by _G. C. Greubel_, Feb 16 2021