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A276369
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G.f. A(x) satisfies: A(x - x*A'(x)) = x^2.
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0
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1, 4, 44, 752, 16816, 454592, 14220544, 501063680, 19533484800, 832009583616, 38365992894464, 1901932780019712, 100804751520935936, 5686686095781380096, 340176827135446794240, 21509238608517787746304, 1433513317187764470480896, 100449987626468857028280320, 7383974661633670900631207936, 568236929745579099067135295488, 45692685682319633001834926309376, 3832507849368533459019374474035200, 334760552511190172092790344295383040
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OFFSET
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2,2
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LINKS
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FORMULA
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G.f. A(x) satisfies:
(1) A(x) = Series_Reversion( x - x*A'(x) )^2.
(2) A(x) = Integral 1 - Series_Reversion( sqrt(A(x)) )/x dx.
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EXAMPLE
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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 +...
such that A(x - x*A'(x)) = x^2.
RELATED SERIES.
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 +...
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 +...
which equals Series_Reversion( x - x*A'(x) ).
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 +...
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PROG
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(PARI) /* From A(x - x*A'(x)) = x^2 */
{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]}
for(n=2, 30, print1(a(n), ", "))
(PARI) /* From A(x) = Series_Reversion( x - x*A'(x) )^2 */
{a(n) = my(A=x^2); for(i=1, n, A = serreverse( x - x*A' +x^2*O(x^n) )^2 ); polcoeff(A, n)}
for(n=2, 30, print1(a(n), ", "))
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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