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.

A144015 Expansion of e.g.f. 1/(1 - sin(4*x))^(1/4).

Original entry on oeis.org

1, 1, 5, 29, 265, 3001, 42125, 696149, 13296145, 287706481, 6959431445, 186061833869, 5448382252825, 173418192216361, 5961442393047965, 220112963745653189, 8687730877758518305, 365023930617143804641, 16266420334783460443685, 766297734521812843642109
Offset: 0

Views

Author

Paul D. Hanna, Sep 09 2008

Keywords

Comments

Row sums of A186492 - Peter Bala, Feb 22 2011.

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 29*x^3/3! + 265*x^4/4! + 3001*x^5/5! +...
log(A(x)) = x + 4*x^2/2! + 16*x^3/3! + 128*x^4/4! + 1280*x^5/5! +...
A(x)^2/A(-x)^2 = 1 + 4*x + 16*x^2/2! + 128*x^3/3! +...+ 4^n*A000111(n)*x^n/n! +...
O.g.f.: 1/(1-x - 4*1*1*x^2/(1-5*x - 4*2*3*x^2/(1-9*x - 4*3*5*x^2/(1-13*x - 4*4*7*x^2/(1-17*x - 4*5*9*x^2/(1-...)))))) [continued fraction by Sergei Gladkovskii].

Crossrefs

Cf. A000111, A001586, A007788, A186492, A230134, A227544, A230114, A235329.
Cf. A007696, A136630.

Programs

  • Mathematica
    CoefficientList[Series[1/(1-Sin[4*x])^(1/4), {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 26 2013 *)
  • PARI
    {a(n)=local(X=x+x*O(x^n)); n!*polcoeff((cos(2*X)-sin(2*X))^(-1/2), n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1+x+x*O(x^n));for(i=0,n,A=exp(intformal(A^2/subst(A^2,x,-x))));n!*polcoeff(A,n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* From A'(x) = A(x)^3 / A(-x)^2: */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^3/subst(A, x, -x)^2 +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    /* 1/sqrt(1-2*Series_Reversion(Integral 1/sqrt(1+4*x-4*x^2) dx)): */
{a(n)=local(A=1);A=1/sqrt(1-2*serreverse(intformal(1/sqrt(1+4*x-4*x^2 +x*O(x^n)))));n!*polcoeff(A, n)}
for(n=0,20,print1(a(n),", "))
  • PARI
    a136630(n, k) = 1/(2^k*k!)*sum(j=0, k, (-1)^(k-j)*(2*j-k)^n*binomial(k, j));
a007696(n) = prod(k=0, n-1, 4*k+1);
a(n) = sum(k=0, n, a007696(k)*(4*I)^(n-k)*a136630(n, k)); \\ Seiichi Manyama, Jun 24 2025

Formula

E.g.f. A(x) satisfies:
(1) A(x) = (cos(2*x) - sin(2*x))^(-1/2).
(2) A(x)^2/A(-x)^2 = 1/cos(4*x) + tan(4*x).
(3) A(x) = exp( Integral A(x)^2/A(-x)^2 dx).
(4) A'(x) = A(x)^3/A(-x)^2 with A(0) = 1.
(5) A(x) = 1/sqrt(1 - 2*Series_Reversion( Integral 1/sqrt(1+4*x-4*x^2) dx )).
G.f.: 1/G(0) where G(k) = 1 - x*(4*k+1) - 4*x^2*(k+1)*(2*k+1)/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 11 2013.
a(n) ~ 2^(3*n+5/4)*n^n/(exp(n)*Pi^(n+1/2)). - Vaclav Kotesovec, Jun 26 2013
a(n) = Sum_{k=0..n} A007696(k) * (4*i)^(n-k) * A136630(n,k), where i is the imaginary unit. - Seiichi Manyama, Jun 24 2025

A234313 E.g.f. satisfies: A'(x) = A(x)^5 * A(-x) with A(0) = 1.

Original entry on oeis.org

1, 1, 4, 34, 376, 5896, 107104, 2445664, 61835776, 1853785216, 60075541504, 2229983878144, 88157067006976, 3901637972801536, 182049480718741504, 9356335870657921024, 503257631887961522176, 29455739077723718189056, 1794347026494847887867904, 117825990265521485020463104
Offset: 0

Views

Author

Paul D. Hanna, Jan 07 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 4*x^2/2! + 34*x^3/3! + 376*x^4/4! + 5896*x^5/5! +...
Related series.
A(x)^5 = 1 + 5*x + 40*x^2/2! + 470*x^3/3! + 7120*x^4/4! + 134000*x^5/5! +...
A(x)^3 = 1 + 3*x + 18*x^2/2! + 180*x^3/3! + 2376*x^4/4! + 40608*x^5/5! +...
Note that 1 - 1/A(x)^3 is an odd function:
1 - 1/A(x)^3 = 3*x + 18*x^3/3! + 1728*x^5/5! + 496368*x^7/7! + 287929728*x^9/9! +...
where Series_Reversion((1 - 1/A(x)^3)/3) = Integral (1-9*x^2)^(1/3) dx.

Links

Crossrefs

Cf. A235329, A235322.

Programs

  • Mathematica
    CoefficientList[1/(1 - 3*InverseSeries[Series[Integrate[(1-9*x^2)^(1/3),x],{x,0,20}],x])^(1/3),x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2014 *)
  • PARI
    {a(n)=local(A=1); for(i=0, n, A=1+intformal(A^5*subst(A, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))
  • PARI
    {a(n)=local(A=1); A=1/(1-3*serreverse(intformal((1-9*x^2 +x*O(x^n))^(1/3))))^(1/3); n!*polcoeff(A, n)}
for(n=0, 20, print1(a(n), ", "))

Formula

E.g.f.: 1/(1 - 3*Series_Reversion( Integral (1-9*x^2)^(1/3) dx ))^(1/3).
Limit n->infinity (a(n)/n!)^(1/n) = 15*GAMMA(5/6) / (sqrt(Pi)*GAMMA(1/3)) = 3.565870639063299... - Vaclav Kotesovec, Jan 28 2014

A235345 E.g.f. satisfies: A'(x) = A(x)^4 / A(-x) with A(0) = 1.

Original entry on oeis.org

1, 1, 5, 37, 425, 6025, 108845, 2276845, 56211665, 1553431825, 48671578325, 1671344339125, 63487049154425, 2605215049488025, 116312045129808125, 5551198271938496125, 284832252600860446625, 15499152225618045102625, 898593034796320550409125, 54905004222557008682573125
Offset: 0

Views

Author

Paul D. Hanna, Jan 06 2014

Keywords

Examples

			E.g.f.: A(x) = 1 + x + 5*x^2/2! + 37*x^3/3! + 425*x^4/4! + 6025*x^5/5! +...
Related series.
A(x)^4 = 1 + 4*x + 32*x^2/2! + 352*x^3/3! + 5120*x^4/4! + 90880*x^5/5! +...
Note that 1 - 1/A(x)^4 is an odd function:
1 - 1/A(x)^4 = 4*x - 32*x^3/3! - 1280*x^5/5! - 235520*x^7/7! - 97894400*x^9/9! +...
where Series_Reversion((1 - 1/A(x)^4)/4) = Integral 1/(1-16*x^2)^(1/4) dx.

Crossrefs

Cf. A235329, A235321.

Programs

  • Mathematica
    CoefficientList[1/(1 - 4*InverseSeries[Series[Integrate[1/(1-16*x^2)^(1/4),x],{x,0,20}],x])^(1/4),x] * Range[0,20]! (* Vaclav Kotesovec, Jan 28 2014 *)
  • PARI
    /* By definition A'(x) = A(x)^4/A(-x): */
{a(n)=local(A=1); for(i=0, n, A=1+intformal(A^4/subst(A, x, -x) +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))
  • PARI
    /* E.g.f. 1/(1 - 4*Series_Reversion(Integral (1-16*x)^(1/4) dx))^(1/4): */
{a(n)=local(A=1); A=1/(1-4*serreverse(intformal(1/(1-4^2*x^2 +x*O(x^n))^(1/4))))^(1/4); n!*polcoeff(A, n)}
for(n=0, 25, print1(a(n), ", "))

Formula

E.g.f.: 1/(1 - 4*Series_Reversion( Integral 1/(1-16*x^2)^(1/4) dx ))^(1/4).
Limit n->infinity (a(n)/n!)^(1/n) = sqrt(2) * GAMMA(1/4)^2 / Pi^(3/2) = 3.33850736669629... - Vaclav Kotesovec, Jan 28 2014
Showing 1-3 of 3 results.

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