OFFSET
0,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..8200
FORMULA
A series quadrisection of A(x) is 2*x^3*A(x^4)^5.
Series bisections of A(x)^2 are:
. (A(x)^2 + A(-x)^2)/2 = A(x^2)^6 + x^2*A(x^2)^4 and
. (A(x)^2 - A(-x)^2)/2 = 2*x*A(x^2)^5.
EXAMPLE
G.f.: A(x) = 1 + x + 3*x^2 + 2*x^3 + 12*x^4 + 7*x^5 + 25*x^6 +...
Related expansions begin:
A(x)^2 = 1 + 2*x + 7*x^2 + 10*x^3 + 37*x^4 + 50*x^5 + 140*x^6 +...
A(x)^3 = 1 + 3*x + 12*x^2 + 25*x^3 + 84*x^4 + 162*x^5 + 444*x^6 +...
A(x)^4 = 1 + 4*x + 18*x^2 + 48*x^3 + 163*x^4 + 388*x^5 + 1098*x^6 +..
A(x)^5 = 1 + 5*x + 25*x^2 + 80*x^3 + 285*x^4 + 786*x^5 + 2340*x^6 +...
A(x)^6 = 1 + 6*x + 33*x^2 + 122*x^3 + 462*x^4 + 1428*x^5 + 4501*x^6 +...
The series bisections of A(x)^2 begin:
. A(x^2)^6 + x^2*A(x^2)^4 = 1 + 7*x^2 + 37*x^4 + 140*x^6 + 510*x^8 + 1591*x^10 +...
. 2*x*A(x^2)^5 = 2*x + 10*x^3 + 50*x^5 + 160*x^7 + 570*x^9 + 1572*x^11 +...
PROG
(PARI) {a(n)=local(A=1+x); for(i=1, 21, A=subst(A, x, x^2+x*O(x^n))^3+x*subst(A, x, x^2+x*O(x^n))^2); polcoeff(A, n)}
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Sep 12 2011
STATUS
approved