A299958 Expansion of root of z^5 + 25*x*z - 1.

1, -5, -25, -125, 0, 13125, 243750, 2921875, 22343750, 0, -3658984375, -77669921875, -1031953125000, -8564355468750, 0, 1584797607421875, 35256063232421875, 487629016113281250, 4190289337158203125, 0, -821167214355468750000, -18710068030548095703125, -264378336959838867187500

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..850

Wikipedia, Lagrange inversion theorem.

Eric Weisstein's World of Mathematics, Quintic Equation.

Formula

25000*(2*n+7)*(4*n-1)*(4*n+9)*(n+1)*a(n)+(n+5)*(n+4)*(n+3)*(n+2)*a(n+5)=0.

a(5*k) = Pochhammer(-1/10, 2*k)*Pochhammer(2/5, 2*k)*(-50000)^k/(Pochhammer(4/5, k)*Pochhammer(3/5, k)*Pochhammer(2/5, k)*k!).

a(5*k+1) = -5*Pochhammer(4/5, 2*k)*Pochhammer(3/10, 2*k)*(-50000)^k/(Pochhammer(6/5, k)*Pochhammer(4/5, k)*Pochhammer(3/5, k)*k!).

a(5*k+2) = -25*Pochhammer(6/5, 2*k)*Pochhammer(7/10, 2*k)*(-50000)^k/(Pochhammer(7/5, k)*Pochhammer(6/5, k)*Pochhammer(4/5, k)*k!).

a(5*k+3) = -125*Pochhammer(8/5, 2*k)*Pochhammer(11/10, 2*k)*(-50000)^k/(Pochhammer(8/5, k)*Pochhammer(7/5, k)*Pochhammer(6/5, k)*k!).

a(5*k+4) = 0.

Maple

f:= gfun:-rectoproc({(25000*(2*n+7))*(4*n-1)*(4*n+9)*(n+1)*a(n)+(n+5)*(n+4)*(n+3)*(n+2)*a(n+5), a(0)=1, a(1)=-5, a(2)=-25, a(3)=-125, a(4)=0}, a(n), remember):
map(f, [$0..40]);

Mathematica

CoefficientList[Root[#^5 + 25*x*# - 1&, 1] + O[x]^40, x] (* Jean-François Alcover, Aug 27 2022 *)

Crossrefs

Sequence in context: A171279 A231636 A099504 * A036156 A097756 A006339

Adjacent sequences: A299955 A299956 A299957 * A299959 A299960 A299961

Keyword

sign

Author

Robert Israel, Feb 22 2018

Status

approved