A380054 G.f. A(x) satisfies: A( A(x)^3 - A(x)^4 ) = x*A(x)^2.

1, 1, 2, 4, 10, 27, 78, 234, 722, 2275, 7288, 23662, 77676, 257372, 859592, 2890838, 9781066, 33271759, 113720380, 390354292, 1345099574, 4651222825, 16134691254, 56132722306, 195807489940, 684712787166, 2399784312560, 8428460653683, 29660054158020, 104564729164209, 369263035193928

Offset: 1

Author

Paul D. Hanna, Jan 27 2025

Keywords

• nonn

Examples

			G.f.: A(x) = x + x^2 + 2*x^3 + 4*x^4 + 10*x^5 + 27*x^6 + 78*x^7 + 234*x^8 + 722*x^9 + 2275*x^10 + 7288*x^11 + 23662*x^12 + ...
where A( A(x)^3 - A(x)^4 ) = x*A(x)^2.
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 5*x^4 + 12*x^5 + 32*x^6 + 90*x^7 + 266*x^8 + 812*x^9 + 2540*x^10 + 8094*x^11 + 26175*x^12 + ...
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 25*x^6 + 72*x^7 + 213*x^8 + 650*x^9 + 2031*x^10 + 6468*x^11 + 20911*x^12 + ...
A(x)^4 = x^4 + 4*x^5 + 14*x^6 + 44*x^7 + 137*x^8 + 428*x^9 + 1356*x^10 + 4356*x^11 + 14172*x^12 + ...
A(x)^3 - A(x)^4 = x^3 + 2*x^4 + 5*x^5 + 11*x^6 + 28*x^7 + 76*x^8 + 222*x^9 + 675*x^10 + 2112*x^11 + 6739*x^12 + ...
where A( A(x)^3 - A(x)^4 ) = x*A(x)^2.
A(x^3 - x^4) = x^3 - x^4 + x^6 - 2*x^7 + x^8 + 2*x^9 - 6*x^10 + 6*x^11 + 2*x^12 - 16*x^13 + 24*x^14 - 6*x^15 - 46*x^16 + 100*x^17 + ...
where A( A(x^3 - x^4)/x^2 ) = x.
SPECIFIC VALUES.
A(t) = 1/2 at t = 0.2678682825020918584942447191444903207470935323567861688...
  where t = 4*A(1/16).
A(t) = 1/3 at t = 0.22799441817101628768717375008489549860256300353023...
  where t = 9*A(2/81).
A(t) = 1/4 at t = 0.18975000755140499291247741716088108267639370901968...
  where t = 16*A(3/256).
A(t) = 1/5 at t = 0.16103727770377462400703035576859030894590037720684...
  where t = 25*A(4/625).
A(1/4) = 0.40005905355209688194814753676993854678408305235431...
A(1/5) = 0.26996802842755752057486556320865534828975851625757...
A(1/6) = 0.20922644319240454092240843641248261969698502660818...
A(1/7) = 0.17175484499974821882568013761069558774534487935356...
A(1/8) = 0.14597605882140622742419332319857860821190968837477...
A(1/16) = 0.06696707062552296462356117978612258018677338308919...
  where A( 4*A(1/16) ) = 1/2.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 1..800

Crossrefs

Cf. A268655.

Programs

PARI

/* From definition A( A(x)^3 - A(x)^4 ) = x*A(x)^2 */
{a(n) = my(A=[1, 1]); for(i=1, n, A = concat(A, 0); G=x*Ser(A); A[#A] = -Vec( subst(G, x, G^3 - G^4 )/G )[#A] ); A[n]}
for(n=1, 40, print1(a(n), ", "))

PARI

/* Using Series Reversion of A(x^3-x^4)/x^2 */
{a(n) = my(A=x); for(i=1, n, A = serreverse( subst(A, x, x^3 - x^4 +x^3*O(x^n))/x^2) ); polcoeff(A, n)}
for(n=1, 40, print1(a(n), ", "))

Formula

G.f. A(x) = Sum_{n>=1} a(n)*x^n satisfies the following formulas.

(1) A( A(x)^3 - A(x)^4 ) = x*A(x)^2.

(2) A( A(x^3 - x^4)/x^2 ) = x.

(3) A( x^3*A(x)^6 - x^4*A(x)^8 ) = x^2*A(x)^7 - x^2*A(x)^8.