A258878 - OEIS

%I #21 Jun 13 2015 23:42:57

%S 1,-2,-20,-3320,-1598960,-1757280800,-3687555924800,

%T -13169930119702400,-73877683147510880000,-613509458527719828800000,

%U -7207218902820454669458560000,-115535941439664355284062432000000,-2454583328787383660694513356633600000,-67459240631654340522067311327301145600000

%N E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^3)^(1/3) dx ), where the constant of integration is zero.

%F E.g.f.: S(x) = Series_Reversion( Sum_{n>=0} A178575(n)*x^(3*n+1)/(3*n+1)! ).

%F E.g.f.: Let C(x) = Sum_{n>=0} a(n)*x^(3*n)/(3*n)! and S(x) = Sum_{n>=0} a(n)*x^(3*n+1)/(3*n+1)! then C(x) and S(x) satisfy:

%F (1) C(x)^3 + S(x)^3 = 1,

%F (2) S'(x) = C(x),

%F (3) C'(x) = -S(x)^2/C(x),

%F (4) C(x)^2 * C'(x) + S(x)^2 * S'(x) = 0,

%F (5) S(x)/C(x) = Integral 1/C(x)^3 dx,

%F (6) S(x)/C(x) = Series_Reversion( Integral 1/(1+x^3) dx ) = Series_Reversion( Sum_{n>=0} (-1)^n * x^(3*n+1)/(3*n+1) ).

%e E.g.f. with offset 0 is C(x) and e.g.f. with offset 1 is S(x) where:

%e C(x) = 1 - 2*x^3/3! - 20*x^6/6! - 3320*x^9/9! - 1598960*x^12/12! - 1757280800*x^15/15! - 3687555924800*x^18/18! -...

%e S(x) = x - 2*x^4/4! - 20*x^7/7! - 3320*x^10/10! - 1598960*x^13/13! - 1757280800*x^16/16! - 3687555924800*x^19/19! -...

%e such that C(x)^3 + S(x)^3 = 1:

%e C(x)^3 = 1 - 6*x^3/3! + 180*x^6/6! - 3240*x^9/9! + 641520*x^12/12! + 455479200*x^15/15! + 798961838400*x^18/18! +...

%e S(x)^3 = 6*x^3/3! - 180*x^6/6! + 3240*x^9/9! - 641520*x^12/12! - 455479200*x^15/15! - 798961838400*x^18/18! -...

%e Related Expansions.

%e (1) The series reversion of S(x) is Integral 1/(1-x^3)^(1/3) dx:

%e Series_Reversion(S(x)) = x + 2*x^4/4! + 160*x^7/7! + 62720*x^10/10! +...

%e 1/(1-x^3)^(1/3) = 1 + 2*x^3/3! + 160*x^6/6! + 62720*x^9/9! + 68992000*x^12/12! + 163235072000*x^15/15! +...+ A178575(n)*x^(3*n)/(3*n)! +...

%e (2) C(x)^2*C'(x) = -S(x)^2*S'(x) = 0, where:

%e C(x)^2*C'(x) = -2*x^2/2! + 60*x^5/5! - 1080*x^8/8! + 213840*x^11/11! + 151826400*x^14/14! + 266320612800*x^17/17! -...

%e S(x)^2*S'(x) = 2*x^2/2! - 60*x^5/5! + 1080*x^8/8! - 213840*x^11/11! - 151826400*x^14/14! - 266320612800*x^17/17! -...

%e (3) d/dx C(x)^2 = -2*S(x)^2, where:

%e C(x)^2 = 1 - 4*x^3/3! + 40*x^6/6! + 80*x^9/9! + 93280*x^12/12! + 60209600*x^15/15! + 83885507200*x^18/18! +...

%e S(x)^2 = 2*x^2/2! - 20*x^5/5! - 40*x^8/8! - 46640*x^11/11! - 30104800*x^14/14! - 41942753600*x^17/17! -...

%e (4) d/dx S(x)/C(x) = 1/C(x)^3:

%e 1/C(x) = 1 + 2*x^3/3! + 100*x^6/6! + 23480*x^9/9! + 15238960*x^12/12! + 21091796000*x^15/15! + 53393583707200*x^18/18! +...

%e 1/C(x)^3 = 1 + 6*x^3/3! + 540*x^6/6! + 184680*x^9/9! + 157600080*x^12/12! +...

%e S(x)/C(x) = x + 6*x^4/4! + 540*x^7/7! + 184680*x^10/10! + 157600080*x^13/13! + 270419925600*x^16/16! +...+  A258880(n)*x^(3*n+1)/(3*n+1)! +...

%e where

%e Series_Reversion(S(x)/C(x)) = x - x^4/4 + x^7/7 - x^10/10 + x^13/13 - x^16/16 +...

%o (PARI) /* E.g.f. Series_Reversion(Integral 1/(1-x^3)^(1/3) dx): */

%o {a(n)=local(S=x,C=1+x); S = serreverse( intformal(  1/(1-x^3 +x*O(x^(3*n)))^(1/3) )); (3*n+1)!*polcoeff(S,3*n+1)}

%o for(n=0,15,print1(a(n),", "))

%o (PARI) /* E.g.f. C(x) with offset 0: */

%o {a(n)=local(S=x,C=1+x);for(i=1,n,S=intformal(C +x*O(x^(3*n)));C=1-intformal(S^2/C +x*O(x^(3*n)));); (3*n)!*polcoeff(C,3*n)}

%o for(n=0,15,print1(a(n),", "))

%o (PARI) /* E.g.f. S(x) with offset 1: */

%o {a(n)=local(S=x,C=1+x);for(i=1,n+1,S=intformal(C +x*O(x^(3*n)));C=1-intformal(S^2/C +x*O(x^(3*n+1)));); (3*n+1)!*polcoeff(S,3*n+1)}

%o for(n=0,15,print1(a(n),", "))

%Y Cf. A178575, A258880.

%K sign

%O 1,2

%A _Paul D. Hanna_, Jun 13 2015