A263532 G.f. A(x) satisfies: A(x) = x + B(A(x))^2 such that B(x) = x + A(B(x))^3, where B(x) is the g.f. of A263533.

1, 1, 2, 7, 32, 165, 920, 5451, 33932, 220127, 1479568, 10259394, 73148986, 534912298, 4003678718, 30621175937, 238991734932, 1901317053794, 15403799348464, 126986224205129, 1064498810334784, 9068572636018981, 78472353869487618, 689421099954372488, 6147113115698137860

Offset

1,3

Links

Paul D. Hanna, Table of n, a(n) for n = 1..300

Formula

G.f. A(x) and B(x) also satisfy:

(1) A( x - B(x)^2 ) = x.

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

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

(4) B( x - x^3 - B(x)^2 ) = x - B(x)^2.

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

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

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

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

Example

G.f.: A(x) = x + x^2 + 2*x^3 + 7*x^4 + 32*x^5 + 165*x^6 + 920*x^7 + 5451*x^8 + 33932*x^9 + 220127*x^10 + 1479568*x^11 + 10259394*x^12 +...
such that A(x) = x + B(A(x))^2 and B(x) = x + A(B(x))^3 where
B(x) = x + x^3 + 3*x^4 + 12*x^5 + 55*x^6 + 276*x^7 + 1488*x^8 + 8499*x^9 + 50925*x^10 + 317841*x^11 + 2055474*x^12 +...
Also
A(x) = x + B(x)^2 + d/dx B(x)^4/2! + d^2/dx^2 B(x)^6/3! + d^3/dx^3 B(x)^8/4! +...
B(x) = x + A(x)^3 + d/dx A(x)^6/2! + d^2/dx^2 A(x)^9/3! + d^3/dx^3 A(x)^12/4! +...
RELATED SERIES.
A(x)^3 = x^3 + 3*x^4 + 9*x^5 + 34*x^6 + 156*x^7 + 804*x^8 + 4469*x^9 + 26307*x^10 + 162306*x^11 + 1042111*x^12 +...
B(x)^2 = x^2 + 2*x^4 + 6*x^5 + 25*x^6 + 116*x^7 + 585*x^8 + 3158*x^9 + 18024*x^10 + 107802*x^11 + 671257*x^12 +...
Also
A(B(x)) = x + x^2 + 3*x^3 + 12*x^4 + 56*x^5 + 291*x^6 + 1634*x^7 + 9738*x^8 + 60887*x^9 + 396259*x^10 + 2669199*x^11 + 18531931*x^12 +...
where A(B( x - x^3 - B(x)^2 )) = x.
And
B(A(x)) = x + x^2 + 3*x^3 + 13*x^4 + 65*x^5 + 356*x^6 + 2090*x^7 + 12963*x^8 + 84090*x^9 + 566495*x^10 + 3943195*x^11 + 28252008*x^12 +...
where B(A( x - x^2 - A(x)^3 )) = x.

Prog

(PARI) {a(n) = my(A=x, B=x); for(i=1, n, A = x + subst(B^2, x, A +x*O(x^n)); B = x + subst(A^3, x, B); ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=x, B=x); for(i=1, n, A = serreverse(x - B^2 +x*O(x^n)); B = serreverse(x - A^3); ); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A263533.

Sequence in context: A268297 A368962 A363562 * A108524 A366400 A226269

Adjacent sequences: A263529 A263530 A263531 * A263533 A263534 A263535

Keyword

nonn

Author

Paul D. Hanna, Nov 03 2015

Status

approved