Bell number generating function
up vote
1
down vote
favorite
I'm finding in various questions/textbooks that for Bell number $B_n$, defined as $$B_n = sum_{kgeq 0}left( begin{array}{rl}n \ k end{array} right) B_k$$ the exponential generating function is known to be $$B(t) = e^{e^t - 1}$$
I can check that this holds by, i.e., expanding it to Taylor series, but how is this expression derived?
combinatorics
asked Nov 13 at 13:05
Nutle
16010
add a comment |
up vote
1
down vote
favorite
I'm finding in various questions/textbooks that for Bell number $B_n$, defined as $$B_n = sum_{kgeq 0}left( begin{array}{rl}n \ k end{array} right) B_k$$ the exponential generating function is known to be $$B(t) = e^{e^t - 1}$$
I can check that this holds by, i.e., expanding it to Taylor series, but how is this expression derived?
combinatorics
asked Nov 13 at 13:05
Nutle
16010
See page 22 in math.upenn.edu/~wilf/gfologyLinked2.pdf
– Robert Z
Nov 13 at 13:09
@RobertZ very useful reference, thanks! Found some more questions answered there :)
– Nutle
Nov 13 at 13:26
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I'm finding in various questions/textbooks that for Bell number $B_n$, defined as $$B_n = sum_{kgeq 0}left( begin{array}{rl}n \ k end{array} right) B_k$$ the exponential generating function is known to be $$B(t) = e^{e^t - 1}$$
I can check that this holds by, i.e., expanding it to Taylor series, but how is this expression derived?
combinatorics
asked Nov 13 at 13:05
Nutle
16010
I'm finding in various questions/textbooks that for Bell number $B_n$, defined as $$B_n = sum_{kgeq 0}left( begin{array}{rl}n \ k end{array} right) B_k$$ the exponential generating function is known to be $$B(t) = e^{e^t - 1}$$
I can check that this holds by, i.e., expanding it to Taylor series, but how is this expression derived?
combinatorics
combinatorics
asked Nov 13 at 13:05
Nutle
16010
asked Nov 13 at 13:05
Nutle
16010
asked Nov 13 at 13:05
Nutle
16010
asked Nov 13 at 13:05
Nutle
16010
asked Nov 13 at 13:05
Nutle
16010
16010
See page 22 in math.upenn.edu/~wilf/gfologyLinked2.pdf
– Robert Z
Nov 13 at 13:09
@RobertZ very useful reference, thanks! Found some more questions answered there :)
– Nutle
Nov 13 at 13:26
add a comment |
See page 22 in math.upenn.edu/~wilf/gfologyLinked2.pdf
– Robert Z
Nov 13 at 13:09
@RobertZ very useful reference, thanks! Found some more questions answered there :)
– Nutle
Nov 13 at 13:26
See page 22 in math.upenn.edu/~wilf/gfologyLinked2.pdf
– Robert Z
Nov 13 at 13:09
See page 22 in math.upenn.edu/~wilf/gfologyLinked2.pdf
– Robert Z
Nov 13 at 13:09
@RobertZ very useful reference, thanks! Found some more questions answered there :)
– Nutle
Nov 13 at 13:26
@RobertZ very useful reference, thanks! Found some more questions answered there :)
– Nutle
Nov 13 at 13:26
add a comment |
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
active
oldest
votes
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%2f2996705%2fbell-number-generating-function%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
See page 22 in math.upenn.edu/~wilf/gfologyLinked2.pdf
– Robert Z
Nov 13 at 13:09
@RobertZ very useful reference, thanks! Found some more questions answered there :)
– Nutle
Nov 13 at 13:26