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

Author

Robert Israel, Feb 22 2018

Keywords

• sign

Links

• Robert Israel, Table of n, a(n) for n = 0..850
• Wikipedia, Lagrange inversion theorem.
• Eric Weisstein's World of Mathematics, Quintic Equation.

Programs

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 *)

PARI

a(n) = (-25)^n*binomial(n/5+1/5, n)/(n+1); \\ Seiichi Manyama, Jun 21 2025

PARI

my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1-25*x)^(1/5))/x) \\ Seiichi Manyama, Jun 21 2025

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.

From Seiichi Manyama, Jun 21 2025: (Start)

a(n) = (-25)^n * binomial(n/5+1/5,n)/(n+1).

G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^3).

G.f.: (1/x) * Series_Reversion(x/(1-25*x)^(1/5)). (End)