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 1-5 of 5 results.

A258880 E.g.f. satisfies: A(x) = Integral 1 + A(x)^3 dx.

Original entry on oeis.org

1, 6, 540, 184680, 157600080, 270419925600, 816984611467200, 3971317527112003200, 29097143353353192480000, 305823675529741700675520000, 4435486895868663971869188480000, 86036822683997062842122964537600000, 2175352015640142857526698650779456000000
Offset: 0

Views

Author

Paul D. Hanna, Jun 13 2015

Keywords

Comments

Note: Sum_{n>=0} (-1)^n*x^(3*n+1)/(3*n+1) = log( (1+x)/(1-x^3)^(1/3) )/2 + Pi*sqrt(3)/18 - atan( (1-2*x)*sqrt(3)/3 )*sqrt(3)/3.

Examples

			E.g.f.: A(x) = x + 6*x^4/4! + 540*x^7/7! + 184680*x^10/10! + 157600080*x^13/13! + 270419925600*x^16/16! +...
where Series_Reversion(A(x)) =  x - x^4/4 + x^7/7 - x^10/10 + x^13/13 - x^16/16 +...

Links

Crossrefs

Cf. A000182, A000831, A258878, A258901, A258925, A258927, A259112, A259113, A258969.

Programs

  • Mathematica
    terms = 13;
A[_] = 0;
Do[A[x_] = Integrate[1 + A[x]^3, x] + O[x]^k // Normal, {k, 1, 3 terms}];
DeleteCases[CoefficientList[A[x], x] Range[0, 3 terms - 2]!, 0] (* Jean-François Alcover, Jul 25 2018 *)
  • PARI
    {a(n) = local(A=x); A = serreverse( sum(m=0,n, (-1)^m * x^(3*m+1)/(3*m+1) ) +O(x^(3*n+2)) ); (3*n+1)!*polcoeff(A,3*n+1)}
for(n=0,20,print1(a(n),", "))
  • PARI
    /* E.g.f. A(x) = Integral 1 + A(x)^3 dx.: */
{a(n) = local(A=x); for(i=1,n+1, A = intformal( 1 + A^3 + O(x^(3*n+2)) )); (3*n+1)!*polcoeff(A,3*n+1)}
for(n=0,20,print1(a(n),", "))

Formula

E.g.f.: Series_Reversion( Integral 1/(1+x^3) dx ).
E.g.f.: Series_Reversion( Sum_{n>=0} (-1)^n * x^(3*n+1)/(3*n+1) ).
a(n) ~ 3^(15*n/2 + 17/4) * n^(3*n+1) / (exp(3*n) * (2*Pi)^(3*n+3/2)). - Vaclav Kotesovec, Jun 15 2015

A178575 Number of permutations of {1,2,...,3n} whose cycle lengths are all divisible by 3.

Original entry on oeis.org

1, 2, 160, 62720, 68992000, 163235072000, 710399033344000, 5129081020743680000, 57096929922918645760000, 927825111247427993600000000, 21095031729321522862489600000000, 648714415740095471067280179200000000, 26246985260844262759382156050432000000000
Offset: 0

Views

Author

Geoffrey Critzer, Dec 23 2010

Keywords

Examples

			a(1) = 2 because we have (123) and (132).

References

  • Herbert S. Wilf, Generatingfunctiontology, page 209

Links

Crossrefs

Cf. A001818, A258878.

Programs

  • Maple
    a:= n-> factorial(3*n)*(mul(1+3*i, i = 1 .. n-1))/(factorial(n)*3^n): seq(a(n), n = 0 .. 11);
  • Mathematica
    Table[(-1)^(n/3) Binomial[-1/3,n/3]n!,{n,0,30,3}]
  • PARI
    v=Vec(serlaplace(1/(1-x^3+O(x^50))^(1/3))); vector(#v\3,n,v[3*n-2])

Formula

a(n) = (-1)^(n/3)*binomial(-1/3,n/3)*n!.
E.g.f.: 1/(1-x^3)^(1/3).
a(n) = ((3*n)!/(n!*3^n))*Product_{i=1..n-1} (1+3*i) (from the Wilf reference).
a(n) ~ (3*n)! / (Gamma(1/3) * n^(2/3)). - Vaclav Kotesovec, Jun 15 2015
D-finite with recurrence: a(n) = (3*n-1)*(3*n-2)^2*a(n-1), a(0)=1. - Georg Fischer, Jul 02 2021 (from the 3rd formula)

A258900 E.g.f.: S(x) = Series_Reversion( Integral 1/(1-x^4)^(1/4) dx ), where the constant of integration is zero.

Original entry on oeis.org

1, -6, -1764, -7700616, -147910405104, -8310698364852576, -1085420895640591777344, -284168646775526186095019136, -134459287943928269154814258953984, -106506405136317713669903020280294647296
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 - 6*x^4/4! - 1764*x^8/8! - 7700616*x^12/12! - 147910405104*x^16/16! -...
S(x) = x - 6*x^5/5! - 1764*x^9/9! - 7700616*x^13/13! - 147910405104*x^17/17! -...
such that C(x)^4 + S(x)^4 = 1:
C(x)^4 = 1 - 24*x^4/4! + 8064*x^8/8! + 2128896*x^12/12! + 52932870144*x^16/16! +...
S(x)^4 = 24*x^4/4! - 8064*x^8/8! - 2128896*x^12/12! - 52932870144*x^16/16! -...
Related Expansions.
(1) The series reversion of S(x) is Integral 1/(1-x^4)^(1/4) dx:
Series_Reversion(S(x)) = x + 6*x^5/5! + 6300*x^9/9! + 56133000*x^13/13! +...
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)! +...
(2) d/dx S(x)/C(x) = 1/C(x)^4:
1/C(x)^4 = 1 + 24*x^4/4! + 32256*x^8/8! + 285272064*x^12/12! +...
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)! +...
where
Series_Reversion(S(x)/C(x)) = x - x^5/5 + x^9/9 - x^13/13 + x^17/17 - x^21/21 +...

Crossrefs

Cf. A258899, A258901, A258878.

Programs

  • PARI
    /* E.g.f. Series_Reversion(Integral 1/(1-x^4)^(1/4) dx): */
{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)}
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^(4*n))); C=1-intformal(S^3/C^2 +x*O(x^(4*n))); ); (4*n)!*polcoeff(C, 4*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^(4*n))); C=1-intformal(S^3/C^2 +x*O(x^(4*n+1))); ); (4*n+1)!*polcoeff(S, 4*n+1)}
for(n=0, 15, print1(a(n), ", "))

Formula

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:
(1) C(x)^4 + S(x)^4 = 1,
(2) S'(x) = C(x),
(3) C'(x) = -S(x)^3/C(x)^2,
(4) C(x)^3 * C'(x) + S(x)^3 * S'(x) = 0,
(5) S(x)/C(x) = Integral 1/C(x)^4 dx,
(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) ).
(7) S(x)^2/C(x)^2 = tan( 2 * Integral S(x)/C(x) dx ).
(8) C(x)^2 + I*S(x)^2 = exp( 2*I * Integral S(x)/C(x) dx ).

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 1-5 of 5 results.

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