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A294475 a(n) = (-1)^n * A294359(n) / (n+1)^2. 2
1, 1, 4, 34, 480, 9806, 265810, 9031672, 369946976, 17763808796, 979237588724, 60980447270606, 4235264029208444, 324666123698241278, 27234289478093347800, 2481803404797989770784, 244176750407604670797512, 25799371668963384500485336, 2913817630601365750195653270, 350339454794537924316740555298, 44680403720910899340653296364594 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

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).

LINKS

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

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.

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, 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.

CROSSREFS

Cf. A294359, A003714.

Sequence in context: A234291 A193099 A208831 * A198976 A156325 A248654

Adjacent sequences:  A294472 A294473 A294474 * A294476 A294477 A294478

KEYWORD

nonn

AUTHOR

Paul D. Hanna, Nov 03 2017

STATUS

approved

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Last modified August 5 07:54 EDT 2021. Contains 346464 sequences. (Running on oeis4.)