This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A215880 #31 Mar 12 2024 02:40:14
%S A215880 1,0,2,0,76,0,9816,0,2731408,0,1327394080,0,998915378880,0,
%T A215880 1076892609950080,0,1575062807865569536,0,3002359129582191616512,0,
%U A215880 7232182908954405203184640,0,21483641086491531479896545280,0,77155816749532379835853517131776,0
%N A215880 Expansion of e.g.f.: sqrt( -LambertW(x)*LambertW(-x)/x^2 ).
%H A215880 Vincenzo Librandi, <a href="/A215880/b215880.txt">Table of n, a(n) for n = 0..200</a>
%F A215880 E.g.f.: exp( Sum_{n>=1} (2*n)^(2*n-1) * x^(2*n)/(2*n)! ).
%F A215880 E.g.f.: sqrt( Chw(x)^2 - Shw(x)^2 ), where
%F A215880 Chw(x) = Sum_{n>=0} (2*n+1)^(2*n-1) * x^(2*n)/(2*n)! and
%F A215880 Shw(x) = Sum_{n>=0} (2*n+2)^(2*n) * x^(2*n+1)/(2*n+1)!.
%F A215880 E.g.f.: sqrt( Chw(x,t)^2 - Shw(x,t)^2 )^(1/t) for |t|>0, where
%F A215880 Chw(x,t) = Sum_{n>=0} t*(2*n+t)^(2*n-1) * x^(2*n)/(2*n)! and
%F A215880 Shw(x,t) = Sum_{n>=0} t*(2*n+t+1)^(2*n) * x^(2*n+1)/(2*n+1)!.
%F A215880 a(n) = Sum_{k=0..n} (-1)^k*C(n,k) * (k + 1/2)^(k-1) * (n-k + 1/2)^(n-k-1) / 4.
%F A215880 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
%e A215880 E.g.f.: A(x) = 1 + 2*x^2/2! + 76*x^4/4! + 9816*x^6/6! + 2731408*x^8/8! + ...
%e A215880 such that A(x) = sqrt( -LambertW(x)*LambertW(-x)/x^2 ) where
%e A215880 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! + ...
%e A215880 Related expansions:
%e A215880 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! + ...
%e A215880 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)! + ...
%t A215880 CoefficientList[Series[Sqrt[-LambertW[x]*LambertW[-x]/x^2], {x, 0, 20}], x]* Range[0, 20]! (* _Vaclav Kotesovec_, Nov 27 2012 *)
%o A215880 (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)}
%o A215880 (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)}
%o A215880 (PARI) /* E.g.f.: sqrt(Chw(x,t)^2 - Shw(x,t)^2)^(1/t) for any |t|>0: */
%o A215880 {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),
%o A215880 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));
%o A215880 n!*polcoeff(sqrt(Chw_t^2-Shw_t^2)^(1/t), n)}
%o A215880 (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)}
%o A215880 (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)}
%o A215880 for(n=0,21,print1(a(n),", "))
%o A215880 (PARI) x='x+O('x^30); Vec(serlaplace(sqrt(-lambertw(x)*lambertw(-x)/ x^2))) \\ _G. C. Greubel_, Feb 19 2018
%o A215880 (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
%Y A215880 Cf. A215881, A215882, A138734, A216143, A215890, A216409.
%K A215880 nonn
%O A215880 0,3
%A A215880 _Paul D. Hanna_, Aug 25 2012