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A113677
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a(n) = (2*n+1)!*(2*n-2)!/(2*((n-1)!)*(n!)^2), n=1,2,... .
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0
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3, 30, 840, 37800, 2328480, 181621440, 17124307200, 1892235945600, 239683219776000, 34226763784012800, 5438943917677670400, 951815185593592320000, 181869917001114101760000
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OFFSET
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1,1
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COMMENTS
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If the sequence is defined by the gamma function, which extends the factorial, then the limit as z approaches zero of a(z) is -1/4 and so a(0)=-1/4 in that sense. - Michael Somos, Sep 02 2006
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LINKS
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FORMULA
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E.g.f., in Maple notation: sum(a(n)*x^n/n!, n=1..infinity)= 1/2*(EllipticK(4*x^(1/2))-2*EllipticE(4*x^(1/2)))/Pi; Integral representation as n-th moment of a positive function on a positive half axis: a(n)=int(x^n*(1/16)*1/Pi^(3/2)/x^(1/2)*exp(-1/32*x)*BesselK(1, 1/32*x), x=0..infinity), n=1, 2, ... . Note that a(0) is not defined.
G.f.: A(x)=y, with A(0)=-1/4, satisfies 0 = 12*y +(1 -32*x)*y' -16*x^2*y''. - Michael Somos, Sep 02 2006
E.g.f.: A(x)=y, with A(0)=-1/4, satisfies 0 = 12*y +(1 -32*x)*y' +(x -16*x^2)*y''. - Michael Somos, Sep 02 2006
E.g.f.: A(x), with A(0)=-1/4, satisfies 0 = 120*(s6*s0 +54*s5*s1 -945*s4*s2 +1050*s3*s3) +(s6*s2 -16*s5*s3 +18*s4*s4) where s0=A(x), s1=s0', s2=s1', ..., s6=s5'. - Michael Somos, Sep 02 2006
a(n) ~ 2^(4*n - 3/2) * n^(n - 1/2) / (sqrt(Pi) * exp(n)). - Vaclav Kotesovec, Jun 26 2022
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PROG
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(PARI) {a(n)=if(n<1, 0, (2*n+1)!*(2*n-2)!/2/(n-1)!/n!^2)} /* Michael Somos, Sep 02 2006 */
(PARI) {a(n)=local(A, B, C); if(n<1, 0, A = sum(k=1, sqrtint(n), 2*x^k^2, 1+x*O(x^n))^2; B = sum(k=1, n, (k%2)*k*x^k/(1-x^(2*k)), x*O(x^n))/A^2; C = sum(k=1, n, 8*x^(2*k)/(1+x^(2*k))^2, 1+x*O(x^n))/A; n!*polcoeff(subst((A-2*C)/4, x, serreverse(B)), n))} /* Michael Somos, Sep 23 2006 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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