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A135457
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a(n) = (2n-1)!! * Sum_{k=0..n-2}(-1)^k/(2k+1).
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3
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0, 3, 10, 91, 684, 8679, 100542, 1664055, 25991640, 532354635, 10455799410, 255542155155, 6044821114500, 171748491958575, 4751436512960550, 153911731348760175, 4874807783839316400, 177334729873063945875
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OFFSET
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1,2
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LINKS
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FORMULA
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a(n) = (-1/4)(Product_{i=1..n}(2i-3))((2n-1)Pi + 2(-1)^n*Sum_{k>=0}k!/ Product_{j=1..k}(2j+2n-1)). - Benoit Cloitre, Dec 15 2007
a(n+3) = 4*a(n+2) + (4n^2+12n+1)*a(n+1) - (8n^2-2)*a(n) with a(1)=0, a(2)=3, a(3)=10. - Benoit Cloitre, Dec 15 2007
a(n+1) = (2n+1)*(a(n) - (-1)^n (2n-3)!!) with a(1)=0. - Cyril Damamme, Jul 16 2015
a(n) = (2^(n-2)*Gamma(n+1/2)*((-1)^n*(Psi(n/2+1/4)-Psi(n/2-1/4))+Pi))/sqrt(Pi). - Peter Luschny, Jul 18 2015
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MAPLE
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a := n -> (2^(n-2)*GAMMA(n+1/2)*((-1)^n*(Psi(n/2+1/4)-Psi(n/2-1/4))+Pi))/sqrt(Pi);
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MATHEMATICA
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FullSimplify[Table[(2^(n-2)*(n-1/2)!*(Pi+2*(-1)^n*LerchPhi[-1, 1, n-1/2]))/Sqrt[Pi], {n, 1, 20}]] (* Vaclav Kotesovec, Oct 11 2013 *)
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PROG
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(PARI) a(n)=round((-1/4)*prod(i=1, n, 2*i-3)*(Pi*(2*n-1)+2*(-1)^n*sum(k=0, 1500, 1.*k!/prod(i=1, k, (2*i+2*n-1)))))
(Magma) I:=[0, 3, 10]; [n le 3 select I[n] else 4*Self(n-1)+(4*n^2-12*n+1)*Self(n-2)-(8*n^2-48*n+70)*Self(n-3): n in [1..30]]; // Vincenzo Librandi, Jul 17 2015
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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Definition replaced by a simplified one by Cyril Damamme, Jul 18 2015
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STATUS
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approved
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