Mixing
Why is this series given by this Taylor expansion?
$begingroup$
So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is:
$$ N(n) = sum_{k=0}^{mathrm{min}(4,n)}2^k ,{4choose k},{nchoose k}, $$
i.e. $N(0) = 1 $,
and $N(1) = 1 + 8 = 9$
etc.
By accident I found that the formula also happens to work for the coefficients for the Taylor expansion of the function:
$$ frac{(1+x)^4}{(1-x)^5}. $$
I am trying to understand whether there is any physical meaning of this function in the context of my graph theory application.
Does anyone know whether this is a known result? Does that function ring a bell?
sequences-and-series graph-theory taylor-expansion generating-functions
edited Feb 2 at 3:57
Alex Ravsky
42.9k32483
asked Jan 31 at 16:40
SuperCiociaSuperCiocia
295213
$endgroup$
add a comment |
$begingroup$
So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is:
$$ N(n) = sum_{k=0}^{mathrm{min}(4,n)}2^k ,{4choose k},{nchoose k}, $$
i.e. $N(0) = 1 $,
and $N(1) = 1 + 8 = 9$
etc.
By accident I found that the formula also happens to work for the coefficients for the Taylor expansion of the function:
$$ frac{(1+x)^4}{(1-x)^5}. $$
I am trying to understand whether there is any physical meaning of this function in the context of my graph theory application.
Does anyone know whether this is a known result? Does that function ring a bell?
sequences-and-series graph-theory taylor-expansion generating-functions
edited Feb 2 at 3:57
Alex Ravsky
42.9k32483
asked Jan 31 at 16:40
SuperCiociaSuperCiocia
295213
$endgroup$
add a comment |
$begingroup$
So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is:
$$ N(n) = sum_{k=0}^{mathrm{min}(4,n)}2^k ,{4choose k},{nchoose k}, $$
i.e. $N(0) = 1 $,
and $N(1) = 1 + 8 = 9$
etc.
By accident I found that the formula also happens to work for the coefficients for the Taylor expansion of the function:
$$ frac{(1+x)^4}{(1-x)^5}. $$
I am trying to understand whether there is any physical meaning of this function in the context of my graph theory application.
Does anyone know whether this is a known result? Does that function ring a bell?
sequences-and-series graph-theory taylor-expansion generating-functions
edited Feb 2 at 3:57
Alex Ravsky
42.9k32483
asked Jan 31 at 16:40
SuperCiociaSuperCiocia
295213
$endgroup$
So I am doing some graph theory stuff and I know that, given $4$ axes that I can walk along in both directions, the number of reached points after $n$ steps is:
$$ N(n) = sum_{k=0}^{mathrm{min}(4,n)}2^k ,{4choose k},{nchoose k}, $$
i.e. $N(0) = 1 $,
and $N(1) = 1 + 8 = 9$
etc.
By accident I found that the formula also happens to work for the coefficients for the Taylor expansion of the function:
$$ frac{(1+x)^4}{(1-x)^5}. $$
I am trying to understand whether there is any physical meaning of this function in the context of my graph theory application.
Does anyone know whether this is a known result? Does that function ring a bell?
sequences-and-series graph-theory taylor-expansion generating-functions
sequences-and-series graph-theory taylor-expansion generating-functions
edited Feb 2 at 3:57
Alex Ravsky
42.9k32483
asked Jan 31 at 16:40
SuperCiociaSuperCiocia
295213
edited Feb 2 at 3:57
Alex Ravsky
42.9k32483
asked Jan 31 at 16:40
SuperCiociaSuperCiocia
295213
edited Feb 2 at 3:57
Alex Ravsky
42.9k32483
edited Feb 2 at 3:57
Alex Ravsky
42.9k32483
edited Feb 2 at 3:57
Alex Ravsky
42.9k32483
42.9k32483
asked Jan 31 at 16:40
SuperCiociaSuperCiocia
295213
asked Jan 31 at 16:40
SuperCiociaSuperCiocia
295213
asked Jan 31 at 16:40
SuperCiociaSuperCiocia
295213
295213
add a comment |
add a comment |
0
active
oldest
votes
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',
autoActivateHeartbeat: false,
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%2f3095112%2fwhy-is-this-series-given-by-this-taylor-expansion%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
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.
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%2f3095112%2fwhy-is-this-series-given-by-this-taylor-expansion%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
