Timeline for Expressions involving Eulerian numbers of the second kind: trying to show $\sum_{m=0}^{n} (-1)^m(m)m!(2n-m-2)!\left\langle\left\langle n\atop m\right\rangle\right\rangle\neq0$ for even $n$.
Current License: CC BY-SA 3.0
12 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 4, 2013 at 19:12 | history | edited | Pietro Majer | CC BY-SA 3.0 |
added 25 characters in body
|
| Aug 13, 2013 at 8:28 | history | edited | Pietro Majer | CC BY-SA 3.0 |
format
|
| Nov 16, 2010 at 10:27 | history | edited | Pietro Majer | CC BY-SA 2.5 |
added 1 characters in body
|
| Nov 15, 2010 at 2:09 | comment | added | Steffen Marcus | Interesting... I was initially expecting the solution to be a generating function argument of some type, so your approach was a pleasant surprise. I've been unsuccessful in trying to naively write the generating function for either $A(n)$ and $B(n)$ as a convolution. In fact, they originate from playing with polynomials in the Lambert function. | |
| Nov 15, 2010 at 0:24 | comment | added | Pietro Majer | You're welcome! I think you can get further information on the sequence $B(n)$ via a GF. One should start by a GF for the $U_n(x)$, say $u(x,t):=\sum_n U_n(x) t^n$ or $u(x,t):=\sum_n U_n(x) t^n/n!$ that satisfies a PDE coming from the recurrence of the $U_n$ (or another convenient variation). Then $b(t):=\int_{-\infty}^0 u(x,t)x/(1-x) dx$ is essentially a GF for the $B(n)$. | |
| Nov 14, 2010 at 23:51 | comment | added | Steffen Marcus | Thanks Pietro! This is very slick and just the sort of thing we've been hoping for... certainly a much more interesting argument than I expected. Thanks also to Mike and Martin for your efforts/feedback. | |
| Nov 14, 2010 at 23:34 | vote | accept | Steffen Marcus | ||
| Nov 14, 2010 at 19:33 | history | edited | Pietro Majer | CC BY-SA 2.5 |
added 42 characters in body
|
| Nov 14, 2010 at 19:01 | history | edited | Pietro Majer | CC BY-SA 2.5 |
added 196 characters in body
|
| Nov 13, 2010 at 23:03 | history | edited | Pietro Majer | CC BY-SA 2.5 |
added 4 characters in body
|
| Nov 13, 2010 at 22:57 | history | edited | Pietro Majer | CC BY-SA 2.5 |
added 234 characters in body; added 3 characters in body; edited body
|
| Nov 13, 2010 at 22:47 | history | answered | Pietro Majer | CC BY-SA 2.5 |