OFFSET
1,3
COMMENTS
Conjecture: a(n) is odd iff n is a power of 2. - Paul D. Hanna, May 19 2026
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..256
EXAMPLE
G.f.: A(x) = x + x^2 - 4*x^3 + 39*x^4 - 580*x^5 + 11480*x^6 - 285116*x^7 + 8617217*x^8 - 311138320*x^9 + ...
Coefficients in the initial iterations of A(x) begin:
[1, 1, -4, 39, -580, 11480, -285116, 8617217, ...];
[1, 2,(-6), 59, -898, 18228, -463816, 14330618, ...];
[1, 3,(-6),(66), -1048, 21932, -572180, 18055088, ...];
[1, 4, -4, (66),(-1100), 23750, -634548, 20415192, ...];
[1, 5, 0, 65, (-1100),(24430), -666940, 21835125, ...];
[1, 6, 6, 69, -1070, (24430),(-679756), 22603642, ...];
[1, 7, 14, 84, -1008, 24038, (-679756),(22919008), ...];
[1, 8, 24, 116, -888, 23492, -671320, (22919008), ...]; ...
where the above coefficients in parenthesis illustrate the property that the coefficients of x^n in A^{n}(x) and in A^{n-1}(x) are equal.
The main diagonal in the above table equals A395842.
PROG
(PARI) \\ Routine returns the n-th iteration of a function F
{IT(n, F) = my(G=x); for(i=1, n, G = subst(F, x, G)); G}
\\ Calculate and print the terms
{a(n) = my(V=[0, 1, 1], A, m); for(i=3, n, V = concat(V, 0); m = #V-2; A=Ser(V);
V[#V] = -polcoef( IT(m+1, A) - IT(m, A), m+1) ); A=Ser(V); polcoef(GF=A, n)}
{upto(n) = a(n); Vec(GF)}
upto(20) \\ program revised by Paul D. Hanna, May 19 2026
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, May 13 2010
EXTENSIONS
Entry revised by Paul D. Hanna, May 19 2026
STATUS
approved