Polynomial sequence

[Orgasmic Troll]

What is the next polynomial in this sequence?

1. xy

2. x³ + xy²

3. 5x³y + xy³

4. 5x⁵ + 18x³y²+xy⁴

5. ??

[mfgoode]

Originally Posted by TravisT

What is the next polynomial in this sequence?

1. xy

2. x³ + xy²

3. 5x³y + xy³

4. 5x⁵ + 18x³y²+xy⁴

5. ??

So whats the answer Travis?
Please give us the method or line of reasoning

Mally

[Orgasmic Troll]

0. f(x) = sec x
1. f'(x) = sec x tan x
2. f''(x) = sec x tan² x + sec³ x
.
.
.

replace sec x with x and tan x with y to get:

1. xy
2. xy² + x³

etc.

I don't have the answer on me right now, but I can dig it up

[jinydu]

Using Mathematica, the next polynomial is:

61(x^5)y + 58(x^3)(y^3) + x(y^5)

[mfgoode]

Originally Posted by TravisT

0. f(x) = sec x
1. f'(x) = sec x tan x
2. f''(x) = sec x tan² x + sec³ x
.
.
.

replace sec x with x and tan x with y to get:

1. xy
2. xy² + x³

etc.

I don't have the answer on me right now, but I can dig it up

Thank you Travis. Very ingenious of you. Its not how far one can go but the method that counts. Thats one more problem to add to my notes. Thats one more type I can put under my hat. :surprised
Mally

[mfgoode]

Originally Posted by jinydu

Using Mathematica, the next polynomial is:

61(x^5)y + 58(x^3)(y^3) + x(y^5)

I am afraid you have jumped a couple of steps ahead and a couple is putting it mildly. On inspection I found your coefficients to be too high
I have had a bad experience with Mathematica and am not surprised at your result.
To quote Travis' original problem which I noted he has given the third term as 5(x^3)y +x(y^3)
The 4th term he has given is 5(x^5) +18 (x^3)(y^2) +x(y^4).
You can take it from there Jinydu.
Altho' 5 is a prime I doubt if a prime as high as 61 should figure so soon if ever!
But I leave it to the experts on the subject to thrash this out.
I for one am satisfied in not going any further unless it is of practicaL value
in some field or the other. Thanks all the same.
Its the good intention and not the result that matters to me.
Mally

[mfgoode]

Jinydu.
I have just read the original post of Travis which I couldnt locate and referred to my notes instead!
He has asked for the 5th in sequence and maybe you may be right after all.
I am very sorry if that is the case and your labour and interest has not gone in Vain.
Mally