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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A081439 Expansion of exp(2*x)*cosh(x/sqrt(1 - x^2)).

Original entry on oeis.org

1, 2, 5, 14, 53, 242, 1505, 10334, 89129, 797090, 8618045, 94186094, 1220350301, 15745031954, 237660317081, 3534411032894, 60889488170321, 1025300949710402, 19847126167227509, 373194859437512654, 8017708459752349061
Offset: 0

Views

Author

Paul Barry, Mar 21 2003

Keywords

Comments

Second binomial transform of expansion of cosh(x/sqrt(1-x^2)).

Links

Crossrefs

Cf. A081440.

Programs

  • Magma
    m:=30; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(2*x)*Cosh(x/Sqrt(1-x^2)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 14 2019
  • Maple
    seq(coeff(series(exp(2*x)*cosh(x/sqrt(1-x^2)), x, n+1)*factorial(n), x, n), n = 0 .. 30); # G. C. Greubel, Aug 14 2019
  • Mathematica
    With[{nn=20}, CoefficientList[Series[Exp[2*x]*Cosh[x/Sqrt[1-x^2]], {x, 0, nn}], x] * Range[0, nn]!] (* Vaclav Kotesovec, Oct 29 2014 *)
  • PARI
    my(x='x+O('x^30)); Vec(serlaplace( exp(2*x)*cosh(x/sqrt(1-x^2)) )) \\ G. C. Greubel, Aug 14 2019
  • Sage
    [factorial(n)*( exp(2*x)*cosh(x/sqrt(1-x^2)) ).series(x,n+1).list()[n] for n in (0..30)] # G. C. Greubel, Aug 14 2019

Formula

D-finite with recurrence: a(n) = 4*a(n-1) + 3*(n-3)*(n-1)*a(n-2) - 6*(n-2)*(2*n-5)*a(n-3) - 3*(n-3)*(n-2)*(n^2 - 7*n + 8)*a(n-4) + 12*(n-4)^2*(n-3)*(n-2)*a(n-5) + (n-5)*(n-4)*(n-3)*(n-2)*(n^2 - 10*n + 12)*a(n-6) - 2*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*(2*n-11)*a(n-7) + 4*(n-7)*(n-6)*(n-5)*(n-4)*(n-3)*(n-2)*a(n-8). - Vaclav Kotesovec, Oct 29 2014

A081442 Expansion of e.g.f.: cosh(x/sqrt(1-x^2)) (even powers).

Original entry on oeis.org

1, 1, 13, 421, 25369, 2449801, 346065061, 67243537453, 17192488230961, 5593309059948049, 2255588021494237501, 1103994926592923677621, 644587811150505183179593, 442516027690815793746696601
Offset: 0

Views

Author

Paul Barry, Mar 21 2003

Keywords

Comments

Periodic zeros suppressed.

Examples

			cosh(x/sqrt(1-x^2)) = 1 + 1/2*x^2 + 13/24*x^4 + 421/720*x^6 + ...

Links

Crossrefs

Cf. A081439, A081440.

Programs

  • Magma
    m:=40; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Cosh(x/Sqrt(1-x^2)) )); [Factorial(2*n-2)*b[2*n-1]: n in [1..Floor((m-2)/2)]]; // G. C. Greubel, Aug 14 2019
  • Maple
    seq(coeff(series(cosh(x/sqrt(1-x^2)), x, 2*n+1)*factorial(2*n), x, 2*n), n = 0 .. 20); # G. C. Greubel, Aug 14 2019
  • Mathematica
    Table[(CoefficientList[Series[Cosh[x/Sqrt[1-x^2]], {x, 0, 40}], x] * Range[0, 40]!)[[n]], {n,1,41,2}] (* Vaclav Kotesovec, Oct 29 2014 *)
  • Maxima
    a(n):=(2*n)!*sum(binomial(n-1,n-j)/(2*j)!,j,0,n); /* Vladimir Kruchinin, May 19 2011 */
  • PARI
    my(x='x+O('x^40)); v=Vec(serlaplace( cosh(x/sqrt(1-x^2)) )); vector(#v\2, n, v[2*n-1]) \\ G. C. Greubel, Aug 14 2019
  • Sage
    [factorial(2*n)*( cosh(x/sqrt(1-x^2)) ).series(x, 2*n+1).list()[2*n] for n in (0..20)] # G. C. Greubel, Aug 14 2019

Formula

a(n) = (2*n)!*Sum_{j=0..n} binomial(n-1,n-j)/(2*j)!. - Vladimir Kruchinin, May 19 2011
E.g.f.: cosh(x/sqrt(1-x^2)) = 1 + x^2/(G(0)-x^2) where G(k)= 2*(2*k+1)*(k+1)*(1-x^2) + x^2 - 2*(2*k+1)*(k+1)*x^2*(1-x^2)/G(k+1); (continued fraction, Euler's 1st kind, 1-step). - Sergei N. Gladkovskii, Aug 06 2012
D-finite with recurrence: a(n) = (12*n^2 - 24*n + 13)*a(n-1) - 12*(n-2)*(n-1)*(2*n-3)^2*a(n-2) + 16*(n-3)*(n-2)^2*(n-1)*(2*n-5)*(2*n-3)*a(n-3). - Vaclav Kotesovec, Oct 29 2014
a(n) ~ 2^(2*n - 1/3) * n^(2*n - 1/3) * exp(3 * 2^(-2/3) * n^(1/3) - 2*n) / sqrt(3) * (1 - 19/72*2^(2/3)/n^(1/3) + 553/5184*2^(1/3)/n^(2/3)). - Vaclav Kotesovec, Oct 29 2014

Extensions

Definition corrected by Joerg Arndt, May 19 2011
Showing 1-2 of 2 results.

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