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FORMULA
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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) ).
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EXAMPLE
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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 +...
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PROG
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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), ", "))
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