Mysterious polynomial sequence
up vote
2
down vote
favorite
Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $nin mathbb{Z}^+$
begin{align}
P(0)&= 1\
P(1)&= a\
P(2)&= a^2+b\
P(3)&= a^3+2ab\
P(4)&= a^4+3a^2b+b^2\
P(5)&= a^5+4a^3b+3ab^2\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
end{align}
More steps upon request.
I'll be grateful for any hints!
sequences-and-series polynomials
add a comment |
up vote
2
down vote
favorite
Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $nin mathbb{Z}^+$
begin{align}
P(0)&= 1\
P(1)&= a\
P(2)&= a^2+b\
P(3)&= a^3+2ab\
P(4)&= a^4+3a^2b+b^2\
P(5)&= a^5+4a^3b+3ab^2\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
end{align}
More steps upon request.
I'll be grateful for any hints!
sequences-and-series polynomials
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $nin mathbb{Z}^+$
begin{align}
P(0)&= 1\
P(1)&= a\
P(2)&= a^2+b\
P(3)&= a^3+2ab\
P(4)&= a^4+3a^2b+b^2\
P(5)&= a^5+4a^3b+3ab^2\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
end{align}
More steps upon request.
I'll be grateful for any hints!
sequences-and-series polynomials
Can someone identify this polynomial sequence? Is it known in mathematics? I'm interested in various properties of this sequence.
I'd like to find $P(n)$, $nin mathbb{Z}^+$
begin{align}
P(0)&= 1\
P(1)&= a\
P(2)&= a^2+b\
P(3)&= a^3+2ab\
P(4)&= a^4+3a^2b+b^2\
P(5)&= a^5+4a^3b+3ab^2\
P(6)&= a^6+5a^4b+6a^2b^2+b^3\
P(7)&= a^7+6 a^5 b+10 a^3 b^2+4 a b^3\
P(8)&= a^8 + 7 a^6 b + 15 a^4 b^2 + 10 a^2 b^3 + b^4\
P(9)&= a^9 + 8 a^7 b + 21 a^5 b^2 + 20 a^3 b^3 + 5 a b^4\
P(10)&= a^{10} + 9 a^8 b + 28 a^6 b^2 + 35 a^4 b^3 + 15 a^2 b^4 + b^5
end{align}
More steps upon request.
I'll be grateful for any hints!
sequences-and-series polynomials
sequences-and-series polynomials
edited 50 mins ago
asked 53 mins ago
Ender
677
677
add a comment |
add a comment |
2 Answers
2
active
oldest
votes
up vote
5
down vote
accepted
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
up vote
1
down vote
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3043962%2fmysterious-polynomial-sequence%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
5
down vote
accepted
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
up vote
5
down vote
accepted
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
up vote
5
down vote
accepted
up vote
5
down vote
accepted
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
Hint. Note that the following recurrence holds: for $ngeq 2$,
$$P(n)=aP(n-1)+bP(n-2).$$
They are related to the Fibonacci polynomials. The wiki page gives a list of properties. For example we have that
$$P(n)=sum_{k=0}^{lfloor n/2rfloor}binom{n-k}{k}a^{n-2k}b^k.$$
edited 23 mins ago
answered 51 mins ago
Robert Z
92.4k1058129
92.4k1058129
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
@BarryCipra Yes it's better to stick to OP's notation. Thanks for pointing out.
– Robert Z
34 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
Many thanks! :) I must study these properties to find if I find something useful
– Ender
18 mins ago
add a comment |
up vote
1
down vote
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
add a comment |
up vote
1
down vote
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
add a comment |
up vote
1
down vote
up vote
1
down vote
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
Try:
$$-frac{2^{-n} left(left(a-sqrt{a^2+4 b}right)^n-left(sqrt{a^2+4
b}+aright)^nright)}{sqrt{a^2+4 b}}$$
answered 43 mins ago
David G. Stork
9,36221232
9,36221232
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3043962%2fmysterious-polynomial-sequence%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown