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A294359
a(n) = [x^n] F(x)^(-(n+1)^2) such that F(x) = F(x^2) + x*F(x^4), where F(x) = Sum_{n>=0} x^A003714(n) and A003714 is the Fibbinary numbers.
2
1, -4, 36, -544, 12000, -353016, 13024690, -578027008, 29965705056, -1776380879600, 118487748235604, -8781184406967264, 715759620936227036, -63634560244855290488, 6127715132571003255000, -635341671628285381320704, 70567080867797749860480968, -8358996420744136578157248864, 1051888164647093035820630830470, -140135781917815169726696222119200, 19704058040921706609228103696785954
OFFSET
0,2
COMMENTS
It is conjectured that all terms are even after the initial '1'.
Fibbinary numbers are integers whose binary representation contains no consecutive ones (see A003714 for definition); it is unexpected that the characteristic function F(x) of the Fibbinary numbers would have only even coefficients of x^n in the negative square powers F(x)^(-(n+1)^2), as described by this sequence.
LINKS
FORMULA
a(n) = (-1)^n * n^2 * A294475(n).
EXAMPLE
Given the characteristic function of the Fibbinary numbers (A003714):
F(x) = 1 + x + x^2 + x^4 + x^5 + x^8 + x^9 + x^10 + x^16 + x^17 + x^18 + x^20 + x^21 + x^32 + x^33 + x^34 + x^36 + x^37 + x^40 + x^41 + x^42 + x^64 + x^65 + x^66 + x^68 + x^69 + x^72 + x^73 + x^74 + x^80 +...+ x^A003714(n) +...
such that F(x) = F(x^2) + x*F(x^4),
then this sequence equals the coefficients of x^n in F(x)^(-(n+1)^2).
ILLUSTRATION OF TERMS.
The table of coefficients of x^k in F(x)^(-n^2) begins:
n=1: [1, -1, 0, 1, -2, 1, 2, -4, 2, 3, -8, 7, 4, -16, 16, 2, -30, ...];
n=2: [1, -4, 6, 0, -19, 40, -26, -56, 166, -160, -110, 560, -705, ...];
n=3: [1, -9, 36, -75, 36, 279, -942, 1278, 531, -5956, 11700, ...];
n=4: [1, -16, 120, -544, 1548, -2192, -2720, 23936, -63426, 67984, ...];
n=5: [1, -25, 300, -2275, 12000, -45005, 112450, -116350, -441375, ...];
n=6: [1, -36, 630, -7104, 57573, -353016, 1668774, -5996664, ...];
n=7: [1, -49, 1176, -18375, 209426, -1846859, 13024690, -74680760, ...];
n=8: [1, -64, 2016, -41600, 631216, -7491392, 72180992, -578027008, ...]; ...
in which the main diagonal forms this sequence.
RELATED SEQUENCES.
Terms (-1)^n * a(n)/(n+1) begin:
[1, 2, 12, 136, 2400, 58836, 1860670, 72253376, 3329522784, 177638087960, ...].
Sequence A294475(n) = (-1)^n * a(n)/(n+1)^2 and begins:
[1, 1, 4, 34, 480, 9806, 265810, 9031672, 369946976, 17763808796, ...].
MATHEMATICA
terms = 21; selfibb = Select[Range[terms], BitAnd[#, 2*#] == 0&]; lenfibb = Length[selfibb]; fibb[0] = 0; fibb[n_] := selfibb[[n]]; F[x_] = Sum[x^fibb[n], {n, 0, lenfibb}]; a[n_] := SeriesCoefficient[F[x]^(-(n + 1)^2), {x, 0, n}]; Array[a, terms, 0] (* Jean-François Alcover, Nov 04 2017 *)
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Nov 03 2017
STATUS
approved