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A215880
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Expansion of e.g.f.: sqrt( -LambertW(x)*LambertW(-x)/x^2 ).
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7
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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
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OFFSET
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0,3
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LINKS
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Vincenzo Librandi, Table of n, a(n) for n = 0..200
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FORMULA
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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
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EXAMPLE
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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)! + ...
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MAPLE
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a:=series(sqrt(-LambertW(x)*LambertW(-x)/x^2), x=0, 26): seq(n!*coeff(a, x, n), n=0..24); # Paolo P. Lava, Mar 27 2019
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MATHEMATICA
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CoefficientList[Series[Sqrt[-LambertW[x]*LambertW[-x]/x^2], {x, 0, 20}], x]* Range[0, 20]! (* Vaclav Kotesovec, Nov 27 2012 *)
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PROG
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(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
(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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CROSSREFS
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Cf. A215881, A215882, A138734, A216143, A215890, A216409.
Sequence in context: A195209 A098276 A335692 * A030129 A012647 A012334
Adjacent sequences: A215877 A215878 A215879 * A215881 A215882 A215883
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna, Aug 25 2012
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STATUS
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approved
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