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%I #2 Mar 30 2012 18:37:16
%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)) = o.g.f. of A157309 have zeros for every other coefficient after initial terms; g.f. of dual sequence A155585 satisfies the same condition.
%C After initial 2 terms, reversing signs yields A157308.
%F Let F(x) = A(x/F(x)) = o.g.f. of A157309, then F(x) satisfies:
%F A(x) = Series_Reversion(x/F(x))/x;
%F A(x) = F(x*A(x));
%F F(x) = A(x/F(x));
%F where A157309 has zeros for every other term after initial [1,1].
%F ...
%F Let G(x) = o.g.f. of A122045, then o.g.f. A(x) satisfies:
%F A(x) = 2+x - x/Series_Reversion(x*G(x));
%F A(x) = 2+x - G( x/(2+x - A(x)) );
%F G(x) = (2 - A(x*G(x))/(1-x);
%F where A122045 is the Euler numbers.
%F ...
%F Let H(x) = o.g.f. of A155585, then o.g.f. A(x) satisfies:
%F A(x) = 2(1+x) - x/Series_Reversion(x*H(x));
%F A(x) = 2 - H( -x/(2 - A(x)) );
%F A(x) = H(-x/A(x));
%F H(x) = A(x*H(-x));
%F where A155585 is defined by e.g.f. exp(x)/cosh(x).
%F ...
%F O.g.f.: A(x) = 2*(1+x) - B(x) where B(x) = g.f. of A157308.
%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 A157309:
%e A157309 = [1,1,0,-1,0,9,0,-176,0,5693,0,-272185,0,...];
%e ...
%e If G(x) = (2 - 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 ...
%e Let H(x) = A(x*H(-x)) = o.g.f. of A155585:
%e A155585 = [1,1,0,-2,0,16,0,-272,0,7936,0,-353792,0,...];
%e ...
%o (PARI) {a(n)=local(A=[1, 1]); for(i=1, n, if(#A%2==1, A=concat(A, 0);); if(#A%2==0, A=concat(A, t); A[ #A]=-subst(Vec(x/serreverse(x*Ser(A)))[ #A], t, 0))); A[n+1]}
%Y Cf. A157308, A157309, A155585, A157304, A157305, A122045 (Euler numbers).
%K sign
%O 0,5
%A _Paul D. Hanna_, Mar 11 2009
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