A173246 Expansion of (1+x)^50 * (1-x)/(1 - x^51).

1, 49, 1175, 18375, 210700, 1888460, 13771940, 83993700, 436994250, 1968555050, 7766844470, 27081460630, 84045912300, 233460867500, 582985137700, 1312983918820, 2672860120455, 4923689695575, 8206149492625, 12352414499425

Offset

0,2

Comments

From G. C. Greubel, Feb 16 2021: (Start)

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.

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)

Links

G. C. Greubel, Table of n, a(n) for n = 0..509

Index entries for linear recurrences with constant coefficients, 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).

Formula

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

From G. C. Greubel, Feb 16 2021: (Start)

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

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

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

Maple

m:= 40;
S:= series( (1+x)^50*(1-x)/(1-x^51), x, m+1);
seq(coeff(S, x, j), j = 0..m); # G. C. Greubel, Feb 16 2021

Mathematica

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

Prog

(Magma)
R<x>:=PowerSeriesRing(Integers(), 40);
Coefficients(R!( (1+x)^50*(1-x)/(1-x^51) )); // G. C. Greubel, Feb 16 2021
(Sage)
def A173246_list(prec):
    P.<x> = PowerSeriesRing(ZZ, prec)
    return P( (1+x)^50*(1-x)/(1-x^51) ).list()
A173246_list(40) # G. C. Greubel, Feb 16 2021

Crossrefs

Cf. A173245.

Sequence in context: A284642 A304046 A264877 * A017765 A163005 A161694

Adjacent sequences: A173243 A173244 A173245 * A173247 A173248 A173249

Keyword

sign,easy,less

Author

Roger L. Bagula, Feb 13 2010

Extensions

Edited by G. C. Greubel, Feb 16 2021

Status

approved