A200820 - OEIS

%I #22 Oct 12 2018 02:11:14

%S 1,1,2,16,120,4296,13056,8004816,-344778624,73663017984,

%T -9783408639744,2198402205799680,-511985055715983360,

%U 152210366371207203840,-51349292670021684664320,20385380417664342515466240

%N Expansion of e.g.f. 1/(1-arctan(1/sqrt(2)*x)^2) (even powers only).

%C This expansion has the form 1/(1-arctan(1/sqrt(2)*x)^2) = Sum_{n>=0} a(2*n)*x^(2*n)/(2n)!.

%H G. C. Greubel, <a href="/A200820/b200820.txt">Table of n, a(n) for n = 0..238</a>

%F a(n) = (2*n)!/2^n*Sum_{k=1..2*n} ((2*k)!*(-1)^((n+k))*Sum_{i=0..2*n-2*k} (2^(i)*Stirling1(i+2*k,2*k)*binomial(2*n-1,i+2*k-1))/(i+2*k)!)), n > 0, a(0)=1.

%F a(n) ~ -(-1)^n * (2*n)! / (n * (log(2*n))^3 * 2^(n-3)) * (1 - 3*(gamma + log(2)) / log(2*n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 11 2018

%p seq(coeff(series(factorial(n)*(1-arctan(1/sqrt(2)*x)^2)^(-1),x,n+1), x, n), n = 0 .. 30,2); # _Muniru A Asiru_, Oct 07 2018

%t Join[{1}, Table[((2*n)!/2^n)*Sum[(2*k)!*(-1)^(n + k)*Sum[(2^j* StirlingS1[j + 2*k, 2*k]*Binomial[2*n - 1, j + 2*k - 1])/(j + 2*k)!, {j, 0, 2*(n - k)}], {k, 1, 2*n}], {n, 1, 50}]] (* _G. C. Greubel_, Oct 05 2018 *)

%o (Maxima)

%o a(n):=if n=0 then 1 else (2*n)!/2^n*sum((2*k)!*(-1)^((n+k))*sum((2^(i)*stirling1(i+2*k,2*k)*binomial(2*n-1,i+2*k-1))/(i+2*k)!,i,0,2*n-2*k),k,1,2*n);

%o (PARI) for(n=0,30, print1(if(n==0,1,((2*n)!/2^n)*sum(k=1,2*n, sum(j=0,2*n-2*k, (2*k)!*(-1)^(n + k)*2^j*stirling(j + 2*k, 2*k, 1)*binomial(2*n - 1, j + 2*k - 1)/(j + 2*k)!) )), ", ")) \\ _G. C. Greubel_, Oct 05 2018

%K sign

%O 0,3

%A _Vladimir Kruchinin_, Nov 23 2011