OFFSET
0,3
COMMENTS
It appears that for k>0, a(k) is odd iff k = 2*A003714(n)+1 for n>=0, where A003714 is the fibbinary numbers (integers whose binary representation contains no consecutive ones); this is true for at least the first 520 terms. [See also A263190 and A263075.] - Paul D. Hanna, Oct 09 2013
Observation of Paul D. Hanna is true for at least the first 1028 terms. - Sean A. Irvine, Apr 25 2014
LINKS
Paul D. Hanna, Table of n, a(n) for n = 0..200
FORMULA
The g.f. A(x) satisfies the following relations.
(1) 0 = [x^(n-1)] A(x)^(n^2), for n > 1.
(2) 0 = [x^(n-1)] A(x)^(n^2) * (1 - n*x*A(x)'/A(x)), for n > 1. - Paul D. Hanna, Oct 22 2020
(3) 0 = [x^n] A(x)^(n^2) * (1 - n*x*A(x)'/A(x)), for n > 0. - Paul D. Hanna, Oct 22 2020
EXAMPLE
G.f.: A(x) = 1 + x - 4*x^2 + 25*x^3 - 194*x^4 + 1603*x^5 +...
The coefficients in the square powers of g.f. A(x) begin:
A^1: [1, 1, -4, 25, -194, 1603, -15264, 122316, ...];
A^4: [1, 4, -10, 56, -427, 3360, -33546, 218880, ...];
A^9: [1, 9, 0, 21, -252, 1701, -25992, -2970, ...];
A^16: [1, 16, 56, 0, -84, -784, -18656, -384896, ...];
A^25: [1, 25, 200, 525, 0, -2695, -38600, -878150, ...];
A^36: [1, 36, 486, 3000, 7821, 0, -101322, -1916352, ...];
A^49: [1, 49, 980, 10241, 58898, 170079, 0, -4515000, ...];
A^64: [1, 64, 1760, 27136, 256048, 1500352, 4979712, 0, ...];
A^81: [1, 81, 2916, 61425, 838026, 7720839, 48097152, 184870512, 0,...]; ...
Note how the coefficient of x^n in A(x)^((n+1)^2) = 0 for n>1.
ALTERNATE RELATION.
The coefficients in A(x)^(n^2) * (1 - n*x*A(x)'/A(x)) begin:
n=1: [1, 0, 4, -50, 582, -6412, 76320, -733896, 13283970, ...];
n=2: [1, 2, 0, -28, 427, -5040, 67092, -547200, 15539502, ...];
n=3: [1, 6, 0, 0, 84, -1134, 25992, 3960, 13172355, ...];
n=4: [1, 12, 28, 0, 0, 196, 9328, 288672, 13426530, ...];
n=5: [1, 20, 120, 210, 0, 0, 7720, 351260, 15775425, ...];
n=6: [1, 30, 324, 1500, 2607, 0, 0, 319392, 17452530, ...];
n=7: [1, 42, 700, 5852, 25242, 48594, 0, 0, 15518020, ...];
n=8: [1, 56, 1320, 16960, 128024, 562632, 1244928, 0, 0, ...];
n=9: [1, 72, 2268, 40950, 465570, 3431484, 16032384, 41082336, 0, 0, ...]; ...
in which the two adjacent diagonals above the main diagonal are all zeros after initial terms, illustrating that
(1) 0 = [x^(n-1)] A(x)^(n^2) * (1 - n*x*A(x)'/A(x)), and
(2) 0 = [x^n] A(x)^(n^2) * (1 - n*x*A(x)'/A(x)), for n > 0.
PROG
(PARI) {a(n) = local(A=[1, 1]); for(m=3, n+1, A=concat(A, 0); A[ #A]=-Vec(Ser(A)^(m^2))[m]/m^2); A[n+1]}
for(n=0, 20, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jan 24 2010
STATUS
approved