This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A258900 #7 Jun 14 2015 14:57:42
%S A258900 1,-6,-1764,-7700616,-147910405104,-8310698364852576,
%T A258900 -1085420895640591777344,-284168646775526186095019136,
%U A258900 -134459287943928269154814258953984,-106506405136317713669903020280294647296
%N A258900 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^4)^(1/4) dx ), where the constant of integration is zero.
%F A258900 Let e.g.f. C(x) = Sum_{n>=0} a(n)*x^(4*n)/(4*n)! and e.g.f. S(x) = Sum_{n>=0} a(n)*x^(4*n+1)/(4*n+1)!, then C(x) and S(x) satisfy:
%F A258900 (1) C(x)^4 + S(x)^4 = 1,
%F A258900 (2) S'(x) = C(x),
%F A258900 (3) C'(x) = -S(x)^3/C(x)^2,
%F A258900 (4) C(x)^3 * C'(x) + S(x)^3 * S'(x) = 0,
%F A258900 (5) S(x)/C(x) = Integral 1/C(x)^4 dx,
%F A258900 (6) S(x)/C(x) = Series_Reversion( Integral 1/(1+x^4) dx ) = Series_Reversion( Sum_{n>=0} (-1)^n * x^(4*n+1)/(4*n+1) ).
%F A258900 (7) S(x)^2/C(x)^2 = tan( 2 * Integral S(x)/C(x) dx ).
%F A258900 (8) C(x)^2 + I*S(x)^2 = exp( 2*I * Integral S(x)/C(x) dx ).
%e A258900 E.g.f. with offset 0 is C(x) and e.g.f. with offset 1 is S(x) where:
%e A258900 C(x) = 1 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! -...
%e A258900 S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! -...
%e A258900 such that C(x)^4 + S(x)^4 = 1:
%e A258900 C(x)^4 = 1 - 24*x^4/4! + 8064*x^8/8! + 2128896*x^12/12! + 52932870144*x^16/16! +...
%e A258900 S(x)^4 = 24*x^4/4! - 8064*x^8/8! - 2128896*x^12/12! - 52932870144*x^16/16! -...
%e A258900 Related Expansions.
%e A258900 (1) The series reversion of S(x) is Integral 1/(1-x^4)^(1/4) dx:
%e A258900 Series_Reversion(S(x)) = x + 6*x^5/5! + 6300*x^9/9! + 56133000*x^13/13! +...
%e A258900 1/(1-x^4)^(1/4) = 1 + 6*x^4/4! + 6300*x^8/8! + 56133000*x^12/12! + 1992160170000*x^16/16! +...+ A258899(n)*x^(4*n)/(4*n)! +...
%e A258900 (2) d/dx S(x)/C(x) = 1/C(x)^4:
%e A258900 1/C(x)^4 = 1 + 24*x^4/4! + 32256*x^8/8! + 285272064*x^12/12! +...
%e A258900 S(x)/C(x) = x + 24*x^5/5! + 32256*x^9/9! + 285272064*x^13/13! + 8967114326016*x^17/17! +...+ A258901(n)*x^(4*n+1)/(4*n+1)! +...
%e A258900 where
%e A258900 Series_Reversion(S(x)/C(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 - x^21/21 +...
%o A258900 (PARI) /* E.g.f. Series_Reversion(Integral 1/(1-x^4)^(1/4) dx): */
%o A258900 {a(n)=local(S=x); S = serreverse( intformal( 1/(1-x^4 +x*O(x^(4*n)))^(1/4) )); (4*n+1)!*polcoeff(S,4*n+1)}
%o A258900 for(n=0,15,print1(a(n),", "))
%o A258900 (PARI) /* E.g.f. C(x) with offset 0: */
%o A258900 {a(n)=local(S=x, C=1+x); for(i=1, n, S=intformal(C +x*O(x^(4*n))); C=1-intformal(S^3/C^2 +x*O(x^(4*n))); ); (4*n)!*polcoeff(C, 4*n)}
%o A258900 for(n=0, 15, print1(a(n), ", "))
%o A258900 (PARI) /* E.g.f. S(x) with offset 1: */
%o A258900 {a(n)=local(S=x, C=1+x); for(i=1, n+1, S=intformal(C +x*O(x^(4*n))); C=1-intformal(S^3/C^2 +x*O(x^(4*n+1))); ); (4*n+1)!*polcoeff(S, 4*n+1)}
%o A258900 for(n=0, 15, print1(a(n), ", "))
%Y A258900 Cf. A258899, A258901, A258878.
%K A258900 sign
%O A258900 0,2
%A A258900 _Paul D. Hanna_, Jun 14 2015