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.

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A258925 E.g.f. satisfies: A(x) = Integral 1 + A(x)^5 dx.

Original entry on oeis.org

1, 120, 3024000, 858574080000, 1226178516326400000, 5912338932461445120000000, 75732595735526211882516480000000, 2195068320271703663798288449536000000000, 128322069958974226301129597680106864640000000000
Offset: 0

Views

Author

Paul D. Hanna, Jun 15 2015

Keywords

Examples

			E.g.f.: A(x) = x + 120*x^6/6! + 3024000*x^11/11! + 858574080000*x^16/16! +...
where Series_Reversion(A(x)) = x - x^6/6 + x^11/11 - x^16/16 + x^21/21 +...
Also, A(x) = S(x)/C(x) where
S(x) = x - 24*x^6/6! - 169344*x^11/11! - 25255286784*x^16/16! - 23089632627769344*x^21/21! +...+ A258924(n)*x^(5*n+1)/(5*n+1)! +...
C(x) = 1 - 24*x^5/5! - 169344*x^10/10! - 25255286784*x^15/15! - 23089632627769344*x^20/20! +...+ A258924(n)*x^(5*n)/(5*n)! +...
such that C(x)^5 + S(x)^5 = 1.

Links

Crossrefs

Cf. A258924, A258880, A258901, A258927, A259112, A259113, A258971.

Programs

  • PARI
    /* E.g.f. Series_Reversion( Integral 1/(1+x^5) dx ): */
{a(n) = local(A=x); A = serreverse( intformal( 1/(1 + x^5 + O(x^(5*n+2))) ) ); (5*n+1)!*polcoeff(A, 5*n+1)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* E.g.f. A(x) = Integral 1 + A(x)^5 dx.: */
{a(n) = local(A=x); for(i=1, n+1, A = intformal( 1 + A^5 + O(x^(5*n+2)) )); (5*n+1)!*polcoeff(A, 5*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^5) dx ).
Let C(x) = S'(x) such that S(x) = Series_Reversion( Integral 1/(1-x^5)^(1/5) dx ) is the e.g.f. of A258924, then e.g.f. A(x) of this sequence satisfies:
(2) A(x) = S(x)/C(x),
(3) A(x) = Integral 1/C(x)^5 dx.
From Vaclav Kotesovec, Jun 18 2015: (Start)
a(n) ~ 5^(5*n+5/2) * ((5-sqrt(5))/8)^(5*n/2+5/8) * (5*n)! * n^(1/4) / (sqrt(2) * Gamma(1/4) * Pi^(5*n+5/4)).
a(n) ~ 5^(5*n+5/2) * sin(Pi/5)^(5*n+5/4) * (5*n)! * n^(1/4) / (sqrt(2) * Gamma(1/4) * Pi^(5*n+5/4)).
(End)

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 1-2 of 2 results.

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