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A166748
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E.g.f.: exp(6*asin(x))
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3
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1, 6, 36, 222, 1440, 9990, 74880, 609390, 5391360, 51798150, 539136000, 6060383550, 73322496000, 951480217350, 13198049280000, 195053444556750, 3061947432960000, 50908949029311750, 894088650424320000
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| exp(6*asin(1/2)) is Aleksandr Gelfond's constant exp(Pi).
This is to clarify that the first formula submitted on my behalf for A166748 is good if the existing A166748 terms are counted starting with the offset 1, as WolframAlpha assumes that, when it guesses the formula. The second formula is OK since it was generated from the "manually" observed (by me) recurrence a(n+2)=(n^2+36)*a(n), a(0)=1, a(1)=6 mentioned in above referenced posting at http://groups.google.com/group/sci.math.symbolic [From Alexander R. Povolotsky (pevnev(AT)juno.com), Oct 29 2009]
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LINKS
| A. R. Povolotsky et al., With regards to OEIS A166748, sci.math.symbolic usenet group, 2009
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FORMULA
| a_n = (2^(n-2)*(3*(-1)^n Gamma(-3*i)*Gamma(3*i)-(-1)^n*Gamma(1/2-3*i)*Gamma(1/2+3*i)+3*Gamma(-3*i)*Gamma(3*i)+Gamma(1/2-3*i)*Gamma(1/2+3*i))*Gamma(n/2-(1/2+3*i))*Gamma(n/2-(1/2-3*i)))/(Gamma(-3*i)*Gamma(3*i)*Gamma(1/2-3*i)* Gamma(1/2+3*i)) (from WolframAlpha, contributed by Alexander Povolotsky, Oct 24 2009)
This can be simplified to a(n)= (3*2^(-3 + n)*gamma(((-1 - 6*i) + n)/2)*gamma(((-1 + 6*i) + n)/2)*((1 + (-1)^n)*csch(3*pi) - (-1 + (-1)^n)*sech(3*pi))*sinh(6*pi))/pi - Alexander Povolotsky, Oct 24 2009
a(n)=3*2^(n-1)*(e^(3*pi)-(-1)^n*exp(-3*pi))*|Gamma(n/2+3i)|^2/pi [From R. J. Mathar and M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 25 2009]
A166748(n)=3*2^(n-1)*(e^(3*pi)-(-1)^n*exp(-3*pi))*|Gamma(n/2+3i)|^2/pi [From M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 25 2009]
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PROG
| (PARI) A166748(n)=round(norm(gamma(n/2+3*I))/Pi*if(n%2, cosh(3*Pi), sinh(3*Pi))*3<<n) \\ [From M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 25 2009]
(PARI) a(n)=polcoeff(exp(6*asin(x)), n)*n!
(PARI) A166748(n)=round(norm(gamma(n/2+3*I))/Pi*if(n%2, cosh(3*Pi), sinh(3*Pi))*3<<n) [From M. F. Hasler (www.univ-ag.fr/~mhasler), Oct 25 2009]
(PARI) a(n)=(1+5*(n%2))*prod(k=0, n\2-1, (2*k+n%2)^2+36) [From Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 28 2009]
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CROSSREFS
| Cf. A166741, A006228, A039661.
Sequence in context: A004319 A129324 A180218 * A200378 A085687 A129327
Adjacent sequences: A166745 A166746 A166747 * A166749 A166750 A166751
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KEYWORD
| nonn
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AUTHOR
| Jaume Oliver Lafont (joliverlafont(AT)gmail.com), Oct 21 2009
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