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%I #15 Sep 08 2022 08:45:29
%S 1,1,-2,-5,12,43,-120,-531,1680,8601,-30240,-172965,665280,4161555,
%T -17297280,-116658675,518918400,3735104625,-17643225600,-134498225925,
%U 670442572800,5380583766075,-28158588057600,-236759435017875,1295295050649600,11364769115001225,-64764752532480000
%N Define an array by Q(m, 0) = 1, Q(m, 1) = 1; Q(m, 2k) = (m - 2k + 1)*Q(m+1, 2k-1) - (2k-1)*Q(m+2, 2k-2), m*Q(m, 2k+1) = (m - 2k)*Q(m+1, 2k) - 2k(m+1)*Q(m+2, 2k-1). Sequence gives Q(0,n).
%D V. van der Noort and N. J. A. Sloane, Paper in preparation, 2007.
%H G. C. Greubel, <a href="/A127137/b127137.txt">Table of n, a(n) for n = 0..500</a>
%F See A127080 for e.g.f..
%F a(n) = (-1)^binomial(n,2)*b(n), where b(2*n) = (2*n)!/n! and b(2*n+1) = 4^n*n!* Sum_{j=0..n} binomial(2*j,j)/8^j. - _G. C. Greubel_, Jan 30 2020
%p seq( (-1)^binomial(n,2)*(`if`(`mod`(n,2)=0, n!/(n/2)!, 2^(n-1)*((n-1)/2)!*add( binomial(2*j,j)/8^j, j=0..((n-1)/2)) ) ), n=0..30); # _G. C. Greubel_, Jan 30 2020
%t Q[0, k_]:= (-1)^Binomial[k, 2]*If[EvenQ[k], k!/(k/2)!, 2^((k-1)/2)*(k)!! Beta[1/2, 1/2, (k+1)/2]/Sqrt[2]]//FullSimplify; Table[Q[0, k], {k, 0, 30}] (* _G. C. Greubel_, Jan 30 2020 *)
%o (PARI) a(n) = (-1)^binomial(n, 2)*if(n%2==0, n!/(n/2)!, 2^(n-1)*((n-1)/2)!*sum( j=0, (n-1)/2, binomial(2*j,j)/8^j));
%o vector(31, n, a(n-1)) \\ _G. C. Greubel_, Jan 30 2020
%o (Magma)
%o function b(n)
%o if n mod 2 eq 0 then return Factorial(n)/Gamma(n/2+1);
%o else return 2^(n-1)*Gamma((n+1)/2)*(&+[Binomial(2*j,j)/8^j: j in [0..((n-1)/2)]]);
%o end if; return b; end function;
%o [Round((-1)^Binomial(n, 2)*b(n)): n in [0..30]]; // _G. C. Greubel_, Jan 30 2020
%o (Sage)
%o @CachedFunction
%o def b(k):
%o if (mod(k,2)==0): return factorial(k)/factorial(k/2)
%o else: return 2^(k-1)*factorial((k-1)/2)*sum(binomial(2*j,j)/8^j for j in (0..(k-1)/2))
%o def a(k): return (-1)^binomial(k, 2)*b(k)
%o [a(n) for n in (0..30)] # _G. C. Greubel_, Jan 30 2020
%Y A001813 interleaved with A090470.
%Y Column 0 of array A127080.
%K sign
%O 0,3
%A _N. J. A. Sloane_, Mar 24 2007
%E Typo in name corrected by _G. C. Greubel_, Jan 30 2020
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