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
Links
- Robert Israel, Table of n, a(n) for n = 0..60
Crossrefs
Programs
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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
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Mathematica
CoefficientList[Series[E^x/Sqrt[2-E^(2*x)]-1, {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Jun 03 2013 *) -
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
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
Extensions
Definition corrected by Robert Israel, May 19 2014