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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.
2
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
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
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Nov 03 2015
STATUS
approved