OFFSET
1,2
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..300
FORMULA
G.f. A(x) = Sum_{n>=0} a(n)*x^n satisfies the following formulas.
(1) A( 2*A(x)^2 - 24*A(x)^3 ) = 2*x^2.
(2) A( sqrt( A(2*x^2 - 24*x^3)/2 ) ) = x.
EXAMPLE
G.f.: A(x) = x + 6*x^2 + 84*x^3 + 1656*x^4 + 35760*x^5 + 829152*x^6 + 20140992*x^7 + 506196864*x^8 + 13052347392*x^9 + 343372202496*x^10 + ...
RELATED SERIES.
2*A(x)^2 - 24*A(x)^3 = 2*x^2 - 24*x^4 - 96*x^6 - 3456*x^8 - 10752*x^10 - 681984*x^12 - 1548288*x^14 - 225607680*x^16 - ...
where A( 2*A(x)^2 - 24*A(x)^3 ) = 2*x^2.
Let B(x) satisfy A(B(x)) = B(A(x)) = x, where
B(x) = x - 6*x^2 - 12*x^3 - 216*x^4 - 336*x^5 - 10656*x^6 - 12096*x^7 - 881280*x^8 - 1781760*x^9 - 90408960*x^10 - ...
then B(x)^2 = A(2*x^2 - 24*x^3)/2 which begins
B(x)^2 = x^2 - 12*x^3 + 12*x^4 - 288*x^5 + 2064*x^6 - 12096*x^7 + 158400*x^8 - 1216512*x^9 + ...
PROG
(PARI) {a(n) = my(V=[1]); for(i=1, n, V=concat(V, 0); A = x*Ser(V); V[#V] = -polcoeff(subst(G=A, x, 2*A^2 - 24*A^3 ), #V+1)/4); V[n]}
for(n=1, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jan 25 2024
STATUS
approved