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Expansion of e.g.f. 1/(1+Pi/4-arctan(2*x+1)).
1

%I #12 Mar 27 2019 09:54:49

%S 1,1,0,-2,8,-16,-112,1968,-16896,55680,1243392,-32546304,427932672,

%T -1824506880,-79446663168,2767039739904,-48592416374784,

%U 243186999164928,17201692341633024,-744898387504988160,16285431143888584704,-90779807638034841600

%N Expansion of e.g.f. 1/(1+Pi/4-arctan(2*x+1)).

%H G. C. Greubel, <a href="/A199043/b199043.txt">Table of n, a(n) for n = 0..425</a>

%F a(n) = n!*Sum_{m=1..n} m!*Sum_{k=0..(n-m)/2} (Sum_{i=0..2*k} (2^(i)* stirling1(i+m,m)*C(2*k+m-1,i+m-1))/(i+m)!))*(-1)^(n+m+k)*C(n-1,2*k+m-1), n>0; a(0)=1.

%p a:=series(1/(1+Pi/4-arctan(2*x+1)),x=0,22): seq(n!*coeff(a,x,n),n=0..21); # _Paolo P. Lava_, Mar 27 2019

%t With[{m = 30}, CoefficientList[Series[1/(1 + Pi/4 - ArcTan[2*x + 1])], {x, 0, m} ], x]*Range[0, m]!] (* _G. C. Greubel_, Feb 19 2019 *)

%o (Maxima) a(n):=if n=0 then 1 else n!*sum(m!*sum((sum((2^(i) *stirling1(i+m,m) *binomial(2*k+m-1,i+m-1))/(i+m)!,i,0,2*k))*(-1)^(n+m+k)*binomial(n-1,2*k+m-1), k,0,(n-m)/2),m,1,n); makelist(a(n),n,1,20);

%o (Sage) m = 30; T = taylor(1/(1+pi/4 -arctan(2*x+1)), x, 0, m); [factorial(n)*T.coefficient(x, n) for n in (0..m)] # _G. C. Greubel_, Feb 19 2019

%K sign

%O 0,4

%A _Vladimir Kruchinin_, Nov 02 2011