A376826 Array read by antidiagonals: T(n,k) = n! * [x^n] exp(2*x + (k/2)*x^2), n >= 0, k >= 0.
1, 1, 2, 1, 2, 4, 1, 2, 5, 8, 1, 2, 6, 14, 16, 1, 2, 7, 20, 43, 32, 1, 2, 8, 26, 76, 142, 64, 1, 2, 9, 32, 115, 312, 499, 128, 1, 2, 10, 38, 160, 542, 1384, 1850, 256, 1, 2, 11, 44, 211, 832, 2809, 6512, 7193, 512, 1, 2, 12, 50, 268, 1182, 4864, 15374, 32400, 29186, 1024
Offset: 0
Examples
Array begins:
======================================================
n\k | 0 1 2 3 4 5 6 7 ...
----+-------------------------------------------------
0 | 1 1 1 1 1 1 1 1 ...
1 | 2 2 2 2 2 2 2 2 ...
2 | 4 5 6 7 8 9 10 11 ...
3 | 8 14 20 26 32 38 44 50 ...
4 | 16 43 76 115 160 211 268 331 ...
5 | 32 142 312 542 832 1182 1592 2062 ...
6 | 64 499 1384 2809 4864 7639 11224 15709 ...
7 | 128 1850 6512 15374 29696 50738 79760 118022 ...
...
Links
- Andrew Howroyd, Table of n, a(n) for n = 0..1325 (first 51 antidiagonals)
- Arvind Ayyer, Hiranya Kishore Dey and Digjoy Paul, How large is the character degree sum compared to the character table sum for a finite group?, arXiv preprint arXiv:2406.06036, [math.RT], 2024.
Crossrefs
Programs
-
PARI
T(n,k) = {sum(i=0, n\2, binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i)!/(2^i*i!))}
Formula
E.g.f. of column k: exp(2*x + k*x^2/2).
Column k is the binomial transform of column k of A359762.
T(n,k) = Sum_{i=0..floor(n/2)} binomial(n,2*i) * 2^(n-2*i) * k^i * (2*i-1)!!.
T(n,k) = Sum_{i=0..floor(n/2)} 2^(n-3*i) * k^i * n! / ((n-2*i)! * i!).