A258924 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^5)^(1/5) dx ), where the constant of integration is zero.
1, -24, -169344, -25255286784, -23089632627769344, -79051067969864491597824, -766667475511149432871084621824, -17578325209217134578862801556544159744, -839197248407269659950832532302025663168118784
Offset: 0
Keywords
Examples
E.g.f. with offset 0 is C(x) and e.g.f. with offset 1 is S(x) where: C(x) = 1 - 24*x^5/5! - 169344*x^10/10! - 25255286784*x^15/15! - 23089632627769344*x^20/20! +... S(x) = x - 24*x^6/6! - 169344*x^11/11! - 25255286784*x^16/16! - 23089632627769344*x^21/21! +... such that C(x)^5 + S(x)^5 = 1: C(x)^5 = 1 - 120*x^5/5! + 604800*x^10/10! + 13208832000*x^15/15! +... S(x)^5 = 120*x^5/5! - 604800*x^10/10! - 13208832000*x^15/15! -... Related Expansions. (1) The series reversion of S(x) is Integral 1/(1-x^5)^(1/5) dx: Series_Reversion(S(x)) = x + 24*x^6/6! + 435456*x^11/11! + 115075344384*x^16/16! +... 1/(1-x^5)^(1/5) = 1 + 24*x^5/5! + 435456*x^10/10! + 115075344384*x^15/15! +... (2) d/dx S(x)/C(x) = 1/C(x)^5: 1/C(x)^5 = 1 + 120*x^5/5! + 3024000*x^10/10! + 858574080000*x^15/15! +... S(x)/C(x) = x + 120*x^6/6! + 3024000*x^11/11! + 858574080000*x^16/16! + 1226178516326400000*x^21/21! +...+ A258925(n)*x^(5*n+1)/(5*n+1)! +... where Series_Reversion(S(x)/C(x)) = x - 1/6*x^6 + 1/11*x^11 - 1/16*x^16 + 1/21*x^21 +...
Programs
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PARI
/* E.g.f. Series_Reversion(Integral 1/(1-x^5)^(1/5) dx): */ {a(n)=local(S=x); S = serreverse( intformal( 1/(1-x^5 +x*O(x^(5*n)))^(1/5) )); (5*n+1)!*polcoeff(S, 5*n+1)} for(n=0, 15, print1(a(n), ", ")) -
PARI
/* E.g.f. C(x) with offset 0: */ {a(n)=local(S=x, C=1+x); for(i=1, n, S=intformal(C +x*O(x^(5*n))); C=1-intformal(S^4/C^3 +x*O(x^(5*n))); ); (5*n)!*polcoeff(C, 5*n)} for(n=0, 15, print1(a(n), ", ")) -
PARI
/* E.g.f. S(x) with offset 1: */ {a(n)=local(S=x, C=1+x); for(i=1, n+1, S=intformal(C +x*O(x^(5*n+1))); C=1-intformal(S^4/C^3 +x*O(x^(5*n+1))); ); (5*n+1)!*polcoeff(S, 5*n+1)} for(n=0, 15, print1(a(n), ", "))
Formula
Let e.g.f. C(x) = Sum_{n>=0} a(n)*x^(5*n)/(5*n)! and e.g.f. S(x) = Sum_{n>=0} a(n)*x^(5*n+1)/(5*n+1)!, then C(x) and S(x) satisfy:
(1) C(x)^5 + S(x)^5 = 1,
(2) S'(x) = C(x),
(3) C'(x) = -S(x)^4/C(x)^3,
(4) C(x)^4 * C'(x) + S(x)^4 * S'(x) = 0,
(5) S(x)/C(x) = Integral 1/C(x)^5 dx,
(6) S(x)/C(x) = Series_Reversion( Integral 1/(1+x^5) dx ) = Series_Reversion( Sum_{n>=0} (-1)^n * x^(5*n+1)/(5*n+1) ).