A294475 a(n) = (-1)^n * A294359(n) / (n+1)^2.

1, 1, 4, 34, 480, 9806, 265810, 9031672, 369946976, 17763808796, 979237588724, 60980447270606, 4235264029208444, 324666123698241278, 27234289478093347800, 2481803404797989770784, 244176750407604670797512, 25799371668963384500485336, 2913817630601365750195653270, 350339454794537924316740555298, 44680403720910899340653296364594

Offset: 0

Author

Paul D. Hanna, Nov 03 2017

Keywords

• nonn

Comments

It is conjectured that all terms are even after the initial '1,1'.

Sequence A294359(n) = [x^n] F(x)^(-(n+1)^2) such that F(x) = Sum_{n>=0} x^A003714(n) where A003714 is the Fibbinary numbers.

Fibbinary numbers are integers whose binary representation contains no consecutive ones (see A003714 for definition).

Examples

			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, divided by (-1)^n*(n+1)^2, forms this sequence:
A = [1, 4/2^2, 36/3^2, 544/4^2, 12000/5^2, 353016/6^2, 13024690/7^2, ...].
RELATED SERIES.
Define g.f. A(x) = Sum_{n>=0} a(n)*x^n, then
A(x) = 1 + x + 4*x^2 + 34*x^3 + 480*x^4 + 9806*x^5 + 265810*x^6 + 9031672*x^7 + 369946976*x^8 + 17763808796*x^9 + 979237588724*x^10 +...
and the logarithm of A(x) begins:
Log(A(x)) = x + 7*x^2/2 + 91*x^3/3 + 1767*x^4/4 + 46181*x^5/5 + 1525351*x^6/6 + 61073419*x^7/7 + 2878197983*x^8/8 + 156203074099*x^9/9 + 9599753499547*x^10/10 +  659252738855129*x^11/11 + 50047297413657447*x^12/12 +...
in which it appears that the coefficients consist entirely of odd integers.
		

Links

• Paul D. Hanna, Table of n, a(n) for n = 0..600

Crossrefs

Cf. A294359, A003714.

Formula

a(n) = (-1)^n * [x^n] F(x)^(-(n+1)^2) / (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.