cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A258901 E.g.f. satisfies: A(x) = Integral 1 + A(x)^4 dx.

Original entry on oeis.org

1, 24, 32256, 285272064, 8967114326016, 735868743566229504, 130778914961055994085376, 44390350317502907443360825344, 26290393222157669992962395876622336, 25377887922329300948014930852183837507584, 37855568618678541873143615775486954119570128896
Offset: 0

Views

Author

Paul D. Hanna, Jun 14 2015

Keywords

Examples

			E.g.f.: A(x) = x + 24*x^5/5! + 32256*x^9/9! + 285272064*x^13/13! + 8967114326016*x^17/17! + 735868743566229504*x^21/21! +...
where Series_Reversion(A(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! - 8310698364852576*x^21/21! +...+ A258900(n)*x^(4*n+1)/(4*n+1)! +...
C(x) = 1 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! - 8310698364852576*x^20/20! +...+ A258900(n)*x^(4*n)/(4*n)! +...
such that C(x)^4 + S(x)^4 = 1.

Links

Crossrefs

Cf. A000182, A000831, A258900, A258880, A258925, A258927, A259112, A259113, A258970.

Programs

  • Mathematica
    nmax=20; Table[CoefficientList[InverseSeries[Series[Integrate[1/(1+x^4),x],{x,0,4*nmax+1}],x],x][[4*n-2]] * (4*n-3)!, {n,1,nmax+1}] (* Vaclav Kotesovec, Jun 18 2015 *)
  • PARI
    /* E.g.f. Series_Reversion( Integral 1/(1+x^4) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^4 + O(x^(4*n+2))) ) ); (4*n+1)!*polcoeff(A,4*n+1)}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* E.g.f. A(x) = Integral 1 + A(x)^4 dx.: */
{a(n) = local(A=x); for(i=1,n+1, A = intformal( 1 + A^4 + O(x^(4*n+2)) )); (4*n+1)!*polcoeff(A,4*n+1)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f. A(x) satisfies:
(1) A(x) = Series_Reversion( Integral 1/(1+x^4) dx ).
(2) A(x) = sqrt( tan( 2 * Integral A(x) dx ) ).
Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^4)^(1/4) dx ) is the e.g.f. of A258900, then e.g.f. A(x) of this sequence satisfies:
(3) A(x) = S(x)/C(x),
(4) A(x) = Integral 1/C(x)^4 dx,
(5) A(x)^2 = S(x)^2/C(x)^2 = tan( 2 * Integral S(x)/C(x) dx ).
a(n) ~ 2^(6*n + 14/3) * (4*n)! * n^(1/3) / (3^(1/3) * Gamma(1/3) * Pi^(4*n + 4/3)). - Vaclav Kotesovec, Jun 18 2015

A258924 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^5)^(1/5) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, -24, -169344, -25255286784, -23089632627769344, -79051067969864491597824, -766667475511149432871084621824, -17578325209217134578862801556544159744, -839197248407269659950832532302025663168118784
Offset: 0

Views

Author

Paul D. Hanna, Jun 15 2015

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 +...

Crossrefs

Cf. A258925 (S/C), A258878, A258900, A258927.

Programs

  • 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) ).

A258926 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^6)^(1/6) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, -120, -21859200, -131273353728000, -6725237593471119360000, -1653993087378574357912780800000, -1405832822961504544259161592168448000000, -3334380558587161259470375739654344298987520000000, -18982929854690021819576777610944622891185796965990400000000
Offset: 0

Views

Author

Paul D. Hanna, Jun 14 2015

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 - 120*x^6/6! - 21859200*x^12/12! - 131273353728000*x^18/18! -...
S(x) = x - 120*x^7/7! - 21859200*x^13/13! - 131273353728000*x^19/19! -...
such that C(x)^6 + S(x)^6 = 1:
C(x)^6 = 1 - 720*x^6/6! + 68428800*x^12/12! + 80406577152000*x^18/18! +...
S(x)^6 = 720*x^6/6! - 68428800*x^12/12! - 80406577152000*x^18/18! -...
Related Expansions.
(1) The series reversion of S(x) is Integral 1/(1-x^6)^(1/6) dx:
Series_Reversion(S(x)) = x + 120*x^7/7! + 46569600*x^13/13! + 449549388288000*x^19/19! +...
1/(1-x^6)^(1/6) = 1 + 120*x^6/6! + 46569600*x^12/12! + 449549388288000*x^18/18! +...
(2) d/dx S(x)/C(x) = 1/C(x)^6:
1/C(x)^6 = 1 + 720*x^6/6! + 410572800*x^12/12! + 4492717498368000*x^18/18! +...
S(x)/C(x) = x + 720*x^7/7! + 410572800*x^13/13! + 4492717498368000*x^19/19! + 348990783113936240640000*x^25/25! +...+ A258927(n)*x^(6*n+1)/(6*n+1)! +...
where
Series_Reversion(S(x)/C(x)) = x - x^7/7 + x^13/13 - x^19/19 + x^25/25 - x^31/31 +...

Crossrefs

Cf. A258927 (S/C), A258878, A258900, A258924.

Programs

  • Mathematica
    nmax = 8; a[n_] := SeriesCoefficient[ InverseSeries[ Integrate[1/(1 - x^6)^(1/6), x] + O[x]^(6nmax+2), x], 6n+1]*(6n+1)!; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Apr 26 2017 *)
  • PARI
    /* E.g.f. Series_Reversion(Integral 1/(1-x^6)^(1/6) dx): */
{a(n)=local(S=x); S = serreverse( intformal(  1/(1-x^6 +x*O(x^(6*n)))^(1/6) )); (6*n+1)!*polcoeff(S, 6*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^(6*n))); C=1-intformal(S^5/C^4 +x*O(x^(6*n))); ); (6*n)!*polcoeff(C,6*n)}
for(n=0, 21, 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^(6*n+1))); C=1-intformal(S^5/C^4 +x*O(x^(6*n+1))); ); (6*n+1)!*polcoeff(S,6*n+1)}
for(n=0, 21, print1(a(n), ", "))

Formula

Let e.g.f. C(x) = Sum_{n>=0} a(n)*x^(6*n)/(6*n)! and e.g.f. S(x) = Sum_{n>=0} a(n)*x^(6*n+1)/(6*n+1)!, then C(x) and S(x) satisfy:
(1) C(x)^6 + S(x)^6 = 1,
(2) S'(x) = C(x),
(3) C'(x) = -S(x)^5/C(x)^4,
(4) C(x)^5 * C'(x) + S(x)^5 * S'(x) = 0,
(5) S(x)/C(x) = Integral 1/C(x)^6 dx,
(6) S(x)/C(x) = Series_Reversion( Integral 1/(1+x^6) dx ) = Series_Reversion( Sum_{n>=0} (-1)^n * x^(6*n+1)/(6*n+1) ).
(7) S(x)^3/C(x)^3 = tan( 3 * Integral S(x)^2/C(x)^2 dx ).
(8) C(x)^3 + I*S(x)^3 = exp( 3*I * Integral S(x)^2/C(x)^2 dx ).
Showing 1-3 of 3 results.

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