A124212 Expansion of e.g.f. exp(x)/sqrt(2-exp(2*x)).

1, 2, 8, 56, 560, 7232, 114368, 2139776, 46223360, 1132124672, 30999600128, 938366468096, 31114518056960, 1121542540992512, 43664751042265088, 1826043989622358016, 81635676596544143360

Offset

0,2

Links

Robert Israel, Table of n, a(n) for n = 0..60

Formula

E.g.f. satisfies: A'(x) = A(x) + A(x)^3 with A(0)=1. [From Paul D. Hanna, Oct 04 2008]

G.f.: 1/G(0) where G(k) = 1 - x*(4*k+2)/( 1 - 2*x*(k+1)/G(k+1) ); (continued fraction ). - Sergei N. Gladkovskii, Mar 23 2013

G.f.: 2/G(0), where G(k)= 1 + 1/(1 - x*(8*k+4)/(x*(8*k+4) - 1 + 4*x*(k+1)/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, May 30 2013

a(n) ~ 2^(n+1/2)*n^n/(log(2)^(n+1/2)*exp(n)). - Vaclav Kotesovec, Jun 03 2013

From Peter Bala, Aug 30 2016: (Start)

a(n) = 1/sqrt(2) * Sum_{k >= 0} (1/8)^k*binomial(2*k,k)*(2*k + 1)^n = 1/sqrt(2) * Sum_{k >= 0} (-1/2)^k*binomial(-1/2,k)*(2*k + 1)^n. Cf. A176785, A124214 and A229558.

a(n) = Sum_{k = 0..n} (1/4)^k*binomial(2*k,k)*A145901(n,k).

a(n) = Sum_{k = 0..n} ( Sum_{i = 0..k} (-1)^(k-i)/4^k* binomial(2*k,k)*binomial(k,i)*(2*i + 1)^n ). (End)

a(n) = 2^n * A014307(n). - Seiichi Manyama, Nov 18 2023

Maple

  N:= 60; # to get a(n) for n <= N
S:= series(exp(x)/sqrt(2-exp(2*x)), x, N+1):
seq(coeff(S, x, j), j=0..N); # Robert Israel, May 19 2014

Mathematica

CoefficientList[Series[E^x/Sqrt[2-E^(2*x)]-1, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 03 2013 *)

Prog

(PARI) {a(n)=local(A=1+x+x*O(x^n)); for(i=0, n, A=1+intformal(A+A^3)); n!*polcoeff(A, n)} \\ Paul D. Hanna, Oct 04 2008

Crossrefs

Cf. A000629, A000984, A014307, A124214, A145901, A176785, A186695, A229558.

Cf. A276371, A365777.

Sequence in context: A277498 A097691 A181939 * A326009 A372160 A349562

Adjacent sequences: A124209 A124210 A124211 * A124213 A124214 A124215

Keyword

nonn,easy

Author

Karol A. Penson, Oct 19 2006

Extensions

Definition corrected by Robert Israel, May 19 2014

Status

approved