A081439 -id:A081439 - cp's OEIS frontend

A081440 Expansion of e.g.f.: exp(x)*cosh(x/sqrt(1 - x^2)).

1, 1, 2, 4, 20, 76, 632, 3424, 38096, 265360, 3682592, 31332544, 520705088, 5232870592, 101265169280, 1173634791424, 25911499036928, 340187621683456, 8436057652027904, 123731966851240960, 3404264757518332928

Offset: 0

Paul Barry, Mar 21 2003

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

G. C. Greubel, Table of n, a(n) for n = 0..445

Cf. A081439, A081442.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!( Exp(x)*Cosh(x/Sqrt(1-x^2)) )); [Factorial(n-1)*b[n]: n in [1..m]]; // G. C. Greubel, Aug 14 2019

seq(coeff(series(exp(x)*cosh(x/sqrt(1-x^2)), x, n+1)*factorial(n), x, n), n = 0 .. 25); # G. C. Greubel, Aug 14 2019

With[{nn=25},CoefficientList[Series[Exp[x]Cosh[x/Sqrt[1-x^2]],{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jun 05 2014 *)

my(x='x+O('x^25)); Vec(serlaplace( exp(x)*cosh(x/sqrt(1-x^2)) )) \\ G. C. Greubel, Aug 14 2019

[factorial(n)*( exp(x)*cosh(x/sqrt(1-x^2)) ).series(x,n+1).list()[n] for n in (0..25)] # G. C. Greubel, Aug 14 2019

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

Definition clarified by Harvey P. Dale, Jun 05 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

Paul Barry, Mar 21 2003

Periodic zeros suppressed.

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

G. C. Greubel, Table of n, a(n) for n = 0..220

Cf. A081439, A081440.

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

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

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 *)

a(n):=(2*n)!*sum(binomial(n-1,n-j)/(2*j)!,j,0,n); /* Vladimir Kruchinin, May 19 2011 */

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

[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

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

Definition corrected by Joerg Arndt, May 19 2011

cp's OEIS Frontend

A081440 Expansion of e.g.f.: exp(x)*cosh(x/sqrt(1 - x^2)).

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