OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
Let B(x) be the series reversion of g.f. A(x), so that A(B(x)) = x, then
(1) 4*B(x)^2 = A( 4*x^2 - 16*x^3 ),
(2) B(4*x^2) = 4*A(x)^2 - 16*A(x)^3,
(3) B(4*B(x)^2) = 4*x^2 - 16*x^3.
a(n) ~ c * d^n / n^(3/2), where d = 9.97390550579080951... and c = 0.0367046946616235... - Vaclav Kotesovec, Aug 28 2017
EXAMPLE
G.f.: A(x) = x + 2*x^2 + 6*x^3 + 48*x^4 + 350*x^5 + 2624*x^6 + 19788*x^7 + 164352*x^8 + 1375222*x^9 + 11676672*x^10 + 100528724*x^11 +...
where A( 4*A(x)^2 - 16*A(x)^3 ) = 4*x^2.
RELATED SERIES.
A(x)^2 = x^2 + 4*x^3 + 16*x^4 + 120*x^5 + 928*x^6 + 7224*x^7 + 56576*x^8 + 472944*x^9 + 4019712*x^10 + 34562904*x^11 + 300781568*x^12 +...
A(x)^3 = x^3 + 6*x^4 + 30*x^5 + 224*x^6 + 1806*x^7 + 14592*x^8 + 118236*x^9 + 1001472*x^10 + 8640726*x^11 + 75300864*x^12 +...
Note that 4*A(x)^2 - 16*A(x)^3 is an even function:
4*A(x)^2 - 16*A(x)^3 = 4*x^2 - 32*x^4 + 128*x^6 - 7168*x^8 + 55296*x^10 - 1687552*x^12 + 41222144*x^14 - 1329070080*x^16 +...
Let B(x) be the series reversion of g.f. A(x) such that A(B(x)) = x, then
B(x) = x - 2*x^2 + 2*x^3 - 28*x^4 + 54*x^5 - 412*x^6 + 2516*x^7 - 20280*x^8 + 133670*x^9 - 888620*x^10 + 6071548*x^11 - 41120904*x^12 +...
such that 4*B(x)^2 = A(4*x^2 - 16*x^3) and B(4*x^2) = 4*A(x)^2 - 16*A(x)^3.
PROG
(PARI) {a(n) = my(A=[1], F=x); for(i=1, n, A=concat(A, 0); F=x*Ser(A); A[#A] = -Vec(subst(F, x, 4*F^2 - 16*F^3))[#A]/8); A[n]}
for(n=1, 30, print1(a(n), ", "))
(PARI) {a(n) = my(A=x); for(i=0, n, A = serreverse(sqrt(subst(A/4, x, 4*x^2 - 16*x^3 + x^2*O(x^n))))); polcoeff(A, n)}
for(n=1, 30, print1(a(n), ", ")) \\ Paul D. Hanna, Aug 11 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Apr 21 2016
STATUS
approved