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%I #21 Nov 04 2017 15:31:56
%S 1,-4,36,-544,12000,-353016,13024690,-578027008,29965705056,
%T -1776380879600,118487748235604,-8781184406967264,715759620936227036,
%U -63634560244855290488,6127715132571003255000,-635341671628285381320704,70567080867797749860480968,-8358996420744136578157248864,1051888164647093035820630830470,-140135781917815169726696222119200,19704058040921706609228103696785954
%N 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.
%C It is conjectured that all terms are even after the initial '1'.
%C 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.
%H Paul D. Hanna, <a href="/A294359/b294359.txt">Table of n, a(n) for n = 0..600</a>
%F a(n) = (-1)^n * n^2 * A294475(n).
%e Given the characteristic function of the Fibbinary numbers (A003714):
%e 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) +...
%e such that F(x) = F(x^2) + x*F(x^4),
%e then this sequence equals the coefficients of x^n in F(x)^(-(n+1)^2).
%e ILLUSTRATION OF TERMS.
%e The table of coefficients of x^k in F(x)^(-n^2) begins:
%e n=1: [1, -1, 0, 1, -2, 1, 2, -4, 2, 3, -8, 7, 4, -16, 16, 2, -30, ...];
%e n=2: [1, -4, 6, 0, -19, 40, -26, -56, 166, -160, -110, 560, -705, ...];
%e n=3: [1, -9, 36, -75, 36, 279, -942, 1278, 531, -5956, 11700, ...];
%e n=4: [1, -16, 120, -544, 1548, -2192, -2720, 23936, -63426, 67984, ...];
%e n=5: [1, -25, 300, -2275, 12000, -45005, 112450, -116350, -441375, ...];
%e n=6: [1, -36, 630, -7104, 57573, -353016, 1668774, -5996664, ...];
%e n=7: [1, -49, 1176, -18375, 209426, -1846859, 13024690, -74680760, ...];
%e n=8: [1, -64, 2016, -41600, 631216, -7491392, 72180992, -578027008, ...]; ...
%e in which the main diagonal forms this sequence.
%e RELATED SEQUENCES.
%e Terms (-1)^n * a(n)/(n+1) begin:
%e [1, 2, 12, 136, 2400, 58836, 1860670, 72253376, 3329522784, 177638087960, ...].
%e Sequence A294475(n) = (-1)^n * a(n)/(n+1)^2 and begins:
%e [1, 1, 4, 34, 480, 9806, 265810, 9031672, 369946976, 17763808796, ...].
%t 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 *)
%Y Cf. A085357, A003714, A294475, A292877.
%K sign
%O 0,2
%A _Paul D. Hanna_, Nov 03 2017
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