A185753 G.f. satisfies: A(x/A(x)) = 1 + sqrt(x - x/A(x)).

1, 1, 1, 3, 15, 102, 861, 8593, 98453, 1269924, 18187062, 286183564, 4907331899, 91082993194, 1819518069135, 38929958186607, 888318740697313, 21536467340324252, 552893064959418966, 14985039828839650746

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Formula

Let G(x) be the g.f. of A185754, then g.f. A(x) satisfies:

(1) x + (A(x) - 1)^2 = G(x),

(2) x * A( G(x) ) = G(x),

(3) G( x/A(x) ) = x.

Example

G.f.: A(x) = 1 + x + x^2 + 3*x^3 + 15*x^4 + 102*x^5 + 861*x^6 +...
RELATED SERIES.
A(x/A(x)) = 1 + x + x^3 + 5*x^4 + 37*x^5 + 329*x^6 + 3415*x^7 + 40328*x^8 + 532749*x^9 + 7777531*x^10 + 124315519*x^11 + ...
x - x/A(x) = x^2 + 2*x^4 + 10*x^5 + 75*x^6 + 668*x^7 + 6929*x^8 + 81684*x^9 + 1076987*x^10 + 15694214*x^11 + 250460767*x^12 + ...
sqrt(x - x/A(x)) = x + x^3 + 5*x^4 + 37*x^5 + 329*x^6 + 3415*x^7 + 40328*x^8 + ...
The g.f. of A185754 begins:
G(x) = x + x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 243*x^6 + 2016*x^7 +...
where
(A(x) - 1)^2 = x^2 + 2*x^3 + 7*x^4 + 36*x^5 + 243*x^6 + 2016*x^7 +...
A(G(x)) = 1 + x + 2*x^2 + 7*x^3 + 36*x^4 + 243*x^5 + 2016*x^6 +...

Prog

(PARI) {a(n) = local(A=x+x^2); for(i=1, n, A = 2*A - x -(x/serreverse(A + x^2*O(x^n)) - 1)^2); polcoeff(x/serreverse(A + x^2*O(x^n)), n)}
for(n=0, 30, print1(a(n), ", "))

Crossrefs

Cf. A185754.

Sequence in context: A265164 A348793 A135903 * A174493 A354122 A318618

Adjacent sequences: A185750 A185751 A185752 * A185754 A185755 A185756

Keyword

nonn

Author

Paul D. Hanna, Feb 02 2011

Status

approved