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

1, 0, 1, 3, 12, 55, 276, 1488, 8499, 50925, 317841, 2055474, 13718079, 94189197, 663693860, 4790024805, 35352118896, 266457599124, 2048794481034, 16055504315982, 128133296727983, 1040680738728060, 8596745138651754, 72191575400906196, 615992556016411446, 5338539526681158456

Offset

1,4

Links

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

Formula

G.f. B(x) and A(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.: 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 +...
such that B(x) = x + A(B(x))^3 and A(x) = x + B(A(x))^2 where
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 +...
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(B, 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(B, n)}
for(n=1, 30, print1(a(n), ", "))

Crossrefs

Cf. A263532.

Sequence in context: A120920 A179487 A350265 * A064314 A362085 A185630

Adjacent sequences: A263530 A263531 A263532 * A263534 A263535 A263536

Keyword

nonn

Author

Paul D. Hanna, Nov 03 2015

Status

approved