A276369 - OEIS

%I #5 Sep 09 2016 12:16:41

%S 1,4,44,752,16816,454592,14220544,501063680,19533484800,832009583616,

%T 38365992894464,1901932780019712,100804751520935936,

%U 5686686095781380096,340176827135446794240,21509238608517787746304,1433513317187764470480896,100449987626468857028280320,7383974661633670900631207936,568236929745579099067135295488,45692685682319633001834926309376,3832507849368533459019374474035200,334760552511190172092790344295383040

%N G.f. A(x) satisfies: A(x - x*A'(x)) = x^2.

%F G.f. A(x) satisfies:

%F (1) A(x) = Series_Reversion( x - x*A'(x) )^2.

%F (2) A(x) = Integral 1 - Series_Reversion( sqrt(A(x)) )/x dx.

%e G.f.: A(x) = x^2 + 4*x^3 + 44*x^4 + 752*x^5 + 16816*x^6 + 454592*x^7 + 14220544*x^8 + 501063680*x^9 + 19533484800*x^10 + 832009583616*x^11 + 38365992894464*x^12 +...

%e such that A(x - x*A'(x)) = x^2.

%e RELATED SERIES.

%e A'(x) = 2*x + 12*x^2 + 176*x^3 + 3760*x^4 + 100896*x^5 + 3182144*x^6 + 113764352*x^7 + 4509573120*x^8 + 195334848000*x^9 + 9152105419776*x^10 +...

%e sqrt(A(x)) = x + 2*x^2 + 20*x^3 + 336*x^4 + 7536*x^5 + 205504*x^6 + 6492096*x^7 + 230905472*x^8 + 9077644544*x^9 + 389501370880*x^10 + 18074816193536*x^11 + 900902386305024*x^12 + 47971267420733440*x^13 + 2717009491995623424*x^14 + 163090418520629886976*x^15 +...

%e which equals Series_Reversion( x - x*A'(x) ).

%e Series_Reversion( sqrt(A(x)) ) = x - 2*x^2 - 12*x^3 - 176*x^4 - 3760*x^5 - 100896*x^6 - 3182144*x^7 - 113764352*x^8 - 4509573120*x^9 - 195334848000*x^10 +...

%o (PARI) /* From A(x - x*A'(x)) = x^2 */

%o {a(n) = my(A=[0,1],G); for(i=0,n, A = concat(A,0); G = x*Ser(A); A[#A] = -polcoeff(subst(G,x, x - x*G'),#A) ); A[n]}

%o for(n=2,30,print1(a(n),", "))

%o (PARI) /* From A(x) = Series_Reversion( x - x*A'(x) )^2 */

%o {a(n) = my(A=x^2); for(i=1,n, A = serreverse( x - x*A' +x^2*O(x^n) )^2 ); polcoeff(A,n)}

%o for(n=2,30,print1(a(n),", "))

%K nonn

%O 2,2

%A _Paul D. Hanna_, Sep 08 2016