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

%I A299958 #32 Jun 21 2025 11:49:34
%S A299958 1,-5,-25,-125,0,13125,243750,2921875,22343750,0,-3658984375,
%T A299958 -77669921875,-1031953125000,-8564355468750,0,1584797607421875,
%U A299958 35256063232421875,487629016113281250,4190289337158203125,0,-821167214355468750000,-18710068030548095703125,-264378336959838867187500
%N A299958 Expansion of root of z^5 + 25*x*z - 1.
%H A299958 Robert Israel, <a href="/A299958/b299958.txt">Table of n, a(n) for n = 0..850</a>
%H A299958 Wikipedia, <a href="https://en.wikipedia.org/wiki/Lagrange_inversion_theorem">Lagrange inversion theorem</a>.
%H A299958 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/QuinticEquation.html">Quintic Equation</a>.
%F A299958 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.
%F A299958 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!).
%F A299958 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!).
%F A299958 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!).
%F A299958 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!).
%F A299958 a(5*k+4) = 0.
%F A299958 From _Seiichi Manyama_, Jun 21 2025: (Start)
%F A299958 a(n) = (-25)^n * binomial(n/5+1/5,n)/(n+1).
%F A299958 G.f. A(x) satisfies A(x) = 1/A(-x/A(x)^3).
%F A299958 G.f.: (1/x) * Series_Reversion(x/(1-25*x)^(1/5)). (End)
%p A299958 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):
%p A299958 map(f, [$0..40]);
%t A299958 CoefficientList[Root[#^5 + 25*x*# - 1&, 1] + O[x]^40, x] (* _Jean-François Alcover_, Aug 27 2022 *)
%o A299958 (PARI) a(n) = (-25)^n*binomial(n/5+1/5, n)/(n+1); \\ _Seiichi Manyama_, Jun 21 2025
%o A299958 (PARI) my(N=30, x='x+O('x^N)); Vec(serreverse(x/(1-25*x)^(1/5))/x) \\ _Seiichi Manyama_, Jun 21 2025
%K A299958 sign
%O A299958 0,2
%A A299958 _Robert Israel_, Feb 22 2018