A155134 G.f. satisfies: A(x) = x + A( A(x)^2 + A(x)^3 ).

1, 1, 3, 11, 46, 207, 979, 4796, 24124, 123862, 646494, 3420055, 18296693, 98817934, 538065697, 2950537937, 16279888522, 90317357884, 503498509274, 2819104999847, 15846101219100, 89386208065828, 505843586002176

Offset

1,3

References

Given g.f. A(x), A(x^2 + x^3) equals the g.f. of A213905. - Paul D. Hanna, Nov 22 2024

Links

Paul D. Hanna, Table of n, a(n) for n = 1..500

Formula

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

Given g.f. A(x), let G(x) = A(x)^2 + A(x)^3, then: A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + G(G(G(G(x)))) +...

From Paul D. Hanna, Nov 22 2024: (Start) G.f. A(x) satisfies

A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) A(x^2 + x^3)^n / n!,

A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) A(x^2 + x^3)^n/x / n! ). (End)

Example

G.f.: A(x) = x + x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 207*x^6 + 979*x^7 + 4796*x^8 + 24124*x^9 + 123862*x^10 + ...
RELATED SERIES.
A(A(x)^2 + A(x)^3) = x^2 + 3*x^3 + 11*x^4 + 46*x^5 + 207*x^6 + 979*x^7 + ...
Series_Reversion(A(x)) = x - A(x^2 + x^3) where
A(x^2 + x^3) = x^2 + x^3 + x^4 + 2*x^5 + 4*x^6 + 9*x^7 + 20*x^8 + 47*x^9 + 112*x^10 + 274*x^11 + 678*x^12 + ... + A213905(n)*x^n + ...
Let G(x) = A(x)^2 + A(x)^3, then iterations of G(x) begin:
G(x) = x^2 + 3*x^3 + 10*x^4 + 40*x^5 + 175*x^6 + 812*x^7 + 3924*x^8 + ...;
G(G(x)) = x^4 + 6*x^5 + 32*x^6 + 167*x^7 + 871*x^8 + 4575*x^9 + ...;
G(G(G(x))) = x^8 + 12*x^9 + 100*x^10 + 718*x^11 + 4773*x^12 + ...;
G(G(G(G(x)))) = x^16 + 24*x^17 + 344*x^18 + 3836*x^19 + ...; ...
where A(x) = x + G(x) + G(G(x)) + G(G(G(x))) + ...
From Paul D. Hanna, Nov 22 2024: (Start)
SPECIFIC VALUES.
A(t) = 1/4 at t = 0.163721803005312172141243746243699244139803252175859...
  where t = 1/4 - A(5/64).
A(t) = 1/5 at t = 0.14929082544565318255001231779199880912414962029963...
  where t = 1/5 - A(6/125).
A(t) = 1/6 at t = 0.13309285167083273104655763601713652013338923799749...
  where t = 1/6 - A(7/216).
A(t) = 1/7 at t = 0.11894786300205336222764000085247689964743470629657...
A(t) = 1/8 at t = 0.10709545601252825509073405671918513540742709693060...
A(1/7) = 0.18561525895081910012559177329721712170174318320749...
A(1/8) = 0.15264181229718973749160148106463213549312900436696...
A(1/9) = 0.13088200905189784510104940736611360541714082945455...
A(1/10) = 0.1149632743531762326478326327192402455216273576487...
(End)

Prog

(PARI) {a(n) = my(A=x +x*O(x^n)); if(n<1, 0, for(i=0, ceil(log(n+1)/log(2)), A = serreverse(x-subst(A, x, x^2 + x^3 +x^2*O(x^n)))); polcoef(A, n))}
for(n=1, 30, print1(a(n), ", "))
(PARI) {Dx(n, F) = my(D=F); for(i=1, n, D=deriv(D)); D}
{a(n) = my(A=x+x^2); for(i=1, n, A = x + sum(m=1, n, Dx(m-1, subst(A, x, x^2 + x^3 +x*O(x^n))^m)/m!) +x*O(x^n)); polcoef(A, n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Nov 22 2024

Crossrefs

Cf. A213905 (A(x^2 + x^3)).

Sequence in context: A151141 A275757 A291090 * A006605 A357233 A306822

Adjacent sequences: A155131 A155132 A155133 * A155135 A155136 A155137

Keyword

nonn

Author

Paul D. Hanna, Jan 20 2009

Status

approved