A215880 Expansion of e.g.f.: sqrt( -LambertW(x)*LambertW(-x)/x^2 ).
1, 0, 2, 0, 76, 0, 9816, 0, 2731408, 0, 1327394080, 0, 998915378880, 0, 1076892609950080, 0, 1575062807865569536, 0, 3002359129582191616512, 0, 7232182908954405203184640, 0, 21483641086491531479896545280, 0, 77155816749532379835853517131776, 0
Offset: 0
Keywords
Examples
E.g.f.: A(x) = 1 + 2*x^2/2! + 76*x^4/4! + 9816*x^6/6! + 2731408*x^8/8! + ... such that A(x) = sqrt( -LambertW(x)*LambertW(-x)/x^2 ) where LambertW(x) = x - 2*x^2/2! + 9*x^3/3! - 64*x^4/4! + 625*x^5/5! - 7776*x^6/6! + 117649*x^7/7! - 2097152*x^8/8! + ... + (-n)^(n-1)*x^n/n! + ... Related expansions: A(x)^2 = 1 + 4*x^2/2! + 176*x^4/4! + 24192*x^6/6! + 6966528*x^8/8! + 3459768320*x^10/10! + ... + A138734(n)*x^n/n! + ... log(A(x)) = 2*x^2/2! + 64*x^4/4! + 7776*x^6/6! + 2097152*x^8/8! + 1000000000*x^10/10! + ... + (2*n)^(2*n-1)*x^(2*n)/(2*n)! + ...
Links
- Vincenzo Librandi, Table of n, a(n) for n = 0..200
Programs
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GAP
List([0..25],n->Sum([0..n],k->(-1)^k*Binomial(n,k)*(k+(1/2))^(k-1)*(n-k+(1/2))^(n-k-1)/4)); # Muniru A Asiru, Feb 19 2018
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Mathematica
CoefficientList[Series[Sqrt[-LambertW[x]*LambertW[-x]/x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *) -
PARI
{a(n)=local(LW=sum(m=1, n+1,-(-1)^m*m^(m-1)*x^m/m!)+x^2*O(x^n)); n!*polcoeff(sqrt(-LW*subst(LW, x, -x)/x^2), n)} -
PARI
{a(n)=local(Chw=sum(m=0, n\2, (2*m+1)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n),Shw=sum(m=0, n\2, (2*m+2)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)); n!*polcoeff(sqrt(Chw^2-Shw^2), n)} -
PARI
/* E.g.f.: sqrt(Chw(x,t)^2 - Shw(x,t)^2)^(1/t) for any |t|>0: */ {a(n)=local(Chw_t=sum(m=0, n\2, t*(2*m+t)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n), Shw_t=sum(m=0, n\2, t*(2*m+t+1)^(2*m)*x^(2*m+1)/(2*m+1)!)+x*O(x^n)); n!*polcoeff(sqrt(Chw_t^2-Shw_t^2)^(1/t), n)} -
PARI
{a(n)=n!*polcoeff(exp(sum(m=1, n, (2*m)^(2*m-1)*x^(2*m)/(2*m)!)+x*O(x^n)), n)} -
PARI
{a(n)=sum(k=0, n, (-1)^k*binomial(n, k)*(k+1/2)^(k-1)*(n-k+1/2)^(n-k-1)/4)} for(n=0,21,print1(a(n),", ")) -
PARI
x='x+O('x^30); Vec(serlaplace(sqrt(-lambertw(x)*lambertw(-x)/ x^2))) \\ G. C. Greubel, Feb 19 2018
Formula
E.g.f.: exp( Sum_{n>=1} (2*n)^(2*n-1) * x^(2*n)/(2*n)! ).
E.g.f.: sqrt( Chw(x)^2 - Shw(x)^2 ), where
Chw(x) = Sum_{n>=0} (2*n+1)^(2*n-1) * x^(2*n)/(2*n)! and
Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.
E.g.f.: sqrt( Chw(x,t)^2 - Shw(x,t)^2 )^(1/t) for |t|>0, where
Chw(x,t) = Sum_{n>=0} t*(2*n+t)^(2*n-1) * x^(2*n)/(2*n)! and
Shw(x,t) = Sum_{n>=0} t*(2*n+t+1)^(2*n) * x^(2*n+1)/(2*n+1)!.
a(n) = Sum_{k=0..n} (-1)^k*C(n,k) * (k + 1/2)^(k-1) * (n-k + 1/2)^(n-k-1) / 4.
If n is even, a(n) ~ exp(1)*sqrt(LambertW(exp(-1)))*n^(n-1) = 1.434430245088497558... * n^(n-1). - Vaclav Kotesovec, Nov 27 2012