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%I #7 Jul 26 2018 18:43:30
%S 1,1,-1,0,3,0,-38,0,947,0,-37394,0,2120190,0,-162980012,0,16330173251,
%T 0,-2070201641498,0,324240251016266,0,-61525045423103316,0,
%U 13913915097436287598,0,-3698477457114061621492,0
%N G.f. A(x) satisfies the condition that both A(x) and F(x) = A(x*F(x)) = g.f. of A155585 have zeros for every other coefficient after initial terms; g.f. of dual sequence A157309 satisfies the same condition.
%C After initial 2 terms, reversing signs yields A157310.
%C Conjecture: a(m) == 1 (mod 2) iff m is a power of 2 or m=0. [_Paul D. Hanna_, Mar 17 2009]
%F Let F(x) = o.g.f. of A155585, then o.g.f. A(x) satisfies:
%F A(x) = x/serreverse(x*F(x));
%F A(x) = 2x + F( -x/(A(x) - 2x) );
%F A(x) = F(x/A(x));
%F F(x) = A(x*F(x));
%F where A155585 is defined by e.g.f. exp(x)/cosh(x).
%F ...
%F Let G(x) = o.g.f. of A122045, then o.g.f. A(x) satisfies:
%F A(x) = x + x/serreverse(x*G(x));
%F A(x) = x + G( x/(A(x) - x) );
%F G(x) = A(x*G(x))/(1+x);
%F where A122045 is the Euler numbers.
%F ...
%F O.g.f.: A(x) = 2*(1+x) - H(x) where H(x) = g.f. of A157310.
%e G.f.: A(x) = 1 + x - x^2 + 3*x^4 - 38*x^6 + 947*x^8 - 37394*x^10 +-...
%e RELATED FUNCTIONS.
%e If F(x) = A(x*F(x)) then F(x) = o.g.f. of A155585:
%e A155585 = [1,1,0,-2,0,16,0,-272,0,7936,0,-353792,0,...];
%e ...
%e If G(x) = A(x*G(x))/(1+x) then G(x) = o.g.f. of A122045:
%e A122045 = [1,0,-1,0,5,0,-61,0,1385,0,-50521,0,2702765,0,...];
%e ...
%t terms = 28;
%t F[x_] = Sum[n! x^n/Product[(1 + 2k x), {k, 1, n}], {n, 0, terms+1}] + O[x]^(terms+1);
%t A[x_] = x/InverseSeries[x F[x]];
%t CoefficientList[A[x], x][[1 ;; terms]] (* _Jean-François Alcover_, Jul 26 2018 *)
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==0, A=concat(A, 0);); if(#A%2==1, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); Vec(x/serreverse(x*Ser(A)))[n+1]}
%Y Cf. A157309, A157310, A157304, A157305, A155585, A122045 (Euler numbers).
%Y Cf. A158119. [_Paul D. Hanna_, Mar 17 2009]
%K sign
%O 0,5
%A _Paul D. Hanna_, Mar 11 2009
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