A268655 G.f. A(x) satisfies: A( A(x)^2 - A(x)^3 ) = x*A(x).

1, 1, 1, 2, 6, 17, 45, 123, 356, 1061, 3193, 9691, 29741, 92228, 288331, 907316, 2871818, 9138352, 29217870, 93813839, 302367222, 977913351, 3172727729, 10323213290, 33677758165, 110135406578, 360982631871, 1185632194394, 3901702645568, 12862978506084, 42477650804971, 140496123308659, 465383254510707, 1543693890316144

Offset

1,4

Comments

Compare g.f. to: C( C(x)^2 - C(x)^4 ) = C(x)^2 and C( C(x) - C(x)^2 ) = C(x) where C(x) = x + C(x)^2 is a g.f. of the Catalan numbers (A000108).

Links

Paul D. Hanna, Table of n, a(n) for n = 1..1030

Formula

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

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

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

(3) A( x^2*A(x)^2 - x^3*A(x)^3 ) = x*A(x)^3 - x*A(x)^4. - Paul D. Hanna, Jan 27 2025

a(n) ~ c * d^n / n^(3/2), where d = 3.469666051910765293806107674644687565... and c = 0.1297182904218030477707168121291985... . - Vaclav Kotesovec, Apr 01 2016

Example

G.f.: A(x) = x + x^2 + x^3 + 2*x^4 + 6*x^5 + 17*x^6 + 45*x^7 + 123*x^8 + 356*x^9 + 1061*x^10 + 3193*x^11 + 9691*x^12 + 29741*x^13 + 92228*x^14 + 288331*x^15 +...
such that A( A(x)^2 - A(x)^3 ) = x*A(x).
RELATED SERIES.
A(x)^2 = x^2 + 2*x^3 + 3*x^4 + 6*x^5 + 17*x^6 + 50*x^7 + 140*x^8 + 394*x^9 + 1152*x^10 + 3464*x^11 + 10541*x^12 + 32320*x^13 + 99973*x^14 +...
A(x)^3 = x^3 + 3*x^4 + 6*x^5 + 13*x^6 + 36*x^7 + 108*x^8 + 316*x^9 + 918*x^10 + 2727*x^11 + 8291*x^12 + 25521*x^13 + 79110*x^14 + 246942*x^15 +...
A(x)^2 - A(x)^3 = x^2 + x^3 + 4*x^6 + 14*x^7 + 32*x^8 + 78*x^9 + 234*x^10 + 737*x^11 + 2250*x^12 + 6799*x^13 + 20863*x^14 + 65056*x^15 +...
The series reversion of g.f. A(x) equals A(x^2-x^3)/x, which begins:
A(x^2-x^3)/x = x - x^2 + x^3 - 2*x^4 + 2*x^5 - 3*x^6 + 5*x^7 - 9*x^8 + 18*x^9 - 38*x^10 + 79*x^11 - 162*x^12 + 330*x^13 - 661*x^14 + 1323*x^15 - 2661*x^16 + 5392*x^17 - 11037*x^18 + 22802*x^19 - 47447*x^20 + 99238*x^21 - 208283*x^22 +...
From Paul D. Hanna, Jan 27 2025: (Start)
SPECIFIC VALUES.
A(t) = 1/3 at t = 0.24013374073214123988983456462489207078012900213444...
  where t = 3*A(2/27).
A(t) = 1/4 at t = 0.19674592924044569625589937333088900641450175035895...
  where t = 4*A(3/64).
A(t) = 1/5 at t = 0.16529543213739406099332848310828359098074695081098...
  where t = 5*A(4/125).
A(t) = 1/6 at t = 0.14218203267706262896303316566462654818462639992381...
A(1/4) = 0.35641415325781383207465959857264799761966460153135...
A(1/5) = 0.25556024199336137440464501886530409917602667263205...
A(1/6) = 0.20205811769271029179938393603280122281452115845908...
A(1/8) = 0.14334739250128661828570393274072214200338351321605...
A(2/27) = 0.08004458024404707996327818820829735692670966737814...
  where A( 3*A(2/27) ) = 1/3.
(End)

Prog

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

Crossrefs

Cf. A380054.

Sequence in context: A065068 A109961 A288029 * A350431 A316591 A222115

Adjacent sequences: A268652 A268653 A268654 * A268656 A268657 A268658

Keyword

nonn

Author

Paul D. Hanna, Apr 01 2016

Status

approved