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%I #14 Oct 16 2016 17:05:17
%S 1,1,-4,56,-1452,58368,-3339220,257978168,-25928865116,3293850854208,
%T -516648260338724,98140914400430840,-22212891796114092556,
%U 5908226316995291448256,-1824964039545848666778100,647939176919565713349368184,-262058639306505158896089030332,119788326167873858048176581362560,-61452317226714509108846922021817924
%N G.f. A(x) satisfies: Series_Reversion( A(x)/(1+x)^2 ) = A(x)/(1-x)^2.
%C Compare to: Series_Reversion( F(x)*(1+x) ) = F(x)*(1-x) when F(x) = x/(1-x^2).
%H Paul D. Hanna, <a href="/A277038/b277038.txt">Table of n, a(n) for n = 1..151</a>
%F G.f. A(x) = Sum_{n>=1} a(n) * x^(2*n-1) also satisfies:
%F (1) A( A(x)/(1-x)^2 ) = x*(1 + A(x)/(1-x)^2 )^2.
%F (2) A( A(x)/(1+x)^2 ) = x*(1 - A(x)/(1+x)^2 )^2.
%e G.f.: A(x) = x + x^3 - 4*x^5 + 56*x^7 - 1452*x^9 + 58368*x^11 - 3339220*x^13 + 257978168*x^15 - 25928865116*x^17 + 3293850854208*x^19 - 516648260338724*x^21 +...
%e such that Series_Reversion( A(x)/(1+x)^2 ) = A(x)/(1-x)^2.
%e RELATED SERIES.
%e A(x)/(1-x)^2 = x + 2*x^2 + 4*x^3 + 6*x^4 + 4*x^5 + 2*x^6 + 56*x^7 + 110*x^8 - 1288*x^9 - 2686*x^10 + 54284*x^11 + 111254*x^12 - 3170996*x^13- 6453246*x^14 + 248242672*x^15 + 502938590*x^16 - 25171230608*x^17 - 50845399806*x^18 + 3217331285204*x^19 + 6485507970214*x^20 +...
%e A( A(x)/(1-x)^2 ) = x + 2*x^2 + 5*x^3 + 12*x^4 + 24*x^5 + 36*x^6 + 52*x^7 + 180*x^8 + 408*x^9 - 2068*x^10 - 7020*x^11 + 99612*x^12 + 311800*x^13 - 5938188*x^14 - 18408156*x^15 + 472421988*x^16 + 1452807992*x^17 - 48432768292*x^18 - 148130779164*x^19 + 6239256281260*x^20 +...
%e Series_Reversion(A(x)) = x - x^3 + 7*x^5 - 100*x^7 + 2367*x^9 - 85825*x^11 + 4517124*x^13 - 328704672*x^15 + 31688182799*x^17 - 3908182388291*x^19 + 599808618273807*x^21 - 112055029500302032*x^23 + 25029715121791017796*x^25 - 6586356648857908264220*x^27 +...
%e Incidentally,
%e sqrt(A(x)/x - 1) = x - 2*x^3 + 26*x^5 - 674*x^7 + 27498*x^9 - 1597090*x^11 + 124852818*x^13 - 12654668930*x^15 + 1616915407314*x^17 - 254633210382402*x^19 + 48505913180978218*x^21 - 11001176427967280994*x^23 + 2930605562417054984890*x^25 - 906290920840955364843298*x^27 +...
%o (PARI) {a(n) = my(A = x +x*O(x^(2*n)));for(i=1,2*n, A = A + (x - subst(A/(1+x +x*O(x^(2*n)))^2,x, A/(1-x +x*O(x^(2*n)))^2))/2); polcoeff(A, 2*n-1)}
%o for(n=1, 20, print1(a(n), ", "))
%Y f. A277177.
%K sign
%O 1,3
%A _Paul D. Hanna_, Oct 02 2016
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