This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A376790 #34 Dec 10 2024 10:06:13 %S A376790 0,0,1,0,9,8,0,36,135,44,0,100,864,1458,208,0,225,3500,15552,14094, %T A376790 912,0,441,10800,95000,258048,130491,3840,0,784,27783,413100,2425000, %U A376790 4174848,1187541,15808 %N A376790 Table T(n,k) read by antidiagonals: T(n,k) (n >=1, k >= 2) is the number of inversions in the radix-k digit reversal permutation of 0, 1, ..., k^n-1. %C A376790 T(n,k) is also the maximum possible number of inversions in the sequence of chips in the (n+1)-st layer of the tree in the stable configuration resulting from directed chip-firing of k^n labeled chips 1, 2, ..., k^n starting at the root of an infinite, rooted directed k-ary tree. See Theorem 6.1 in Inagaki, Khovanova, and Luo (2024). %H A376790 David M. W. Evans, <a href="https://doi.org/10.1109/TASSP.1987.1165252">An Improved Digit-Reversal Permutation Algorithm for the Fast Fourier and Hartley Transforms</a>, IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. ASSP-35, No. 8 (1987), 1120-1125. %H A376790 Ryota Inagaki, Tanya Khovanova, and Austin Luo, <a href="https://arxiv.org/abs/2410.23265">Chip Firing on Directed k-ary Trees</a>, arXiv:2410.23265 [math.CO], 2024. %H A376790 Wikipedia, <a href="https://en.wikipedia.org/wiki/Bit-reversal_permutation">Bit-reversal Permutation</a> %F A376790 T(n, k) = (k^(2*n) - n*k^(n+1) + (n-1)*k^n) / 4. %e A376790 For n=3, k=2, the radix-2 digit reversal permutation of 0, 1, 2, ..., 2^3-1 is 0, 4, 2, 6, 1, 5, 3, 7. This permutation has 8 inversions. %e A376790 Array begins: %e A376790 ======================================================= %e A376790 n/k | 2 3 4 5 6 %e A376790 ----+-------------------------------------------------- %e A376790 1 | 0 0 0 0 0 ... %e A376790 2 | 1 9 36 100 225 ... %e A376790 3 | 8 135 864 3500 10800 ... %e A376790 4 | 44 1458 15552 95000 413100 ... %e A376790 5 | 208 14094 258048 2425000 15066000 ... %e A376790 6 | 912 130491 4174848 60937500 543834000 ... %e A376790 7 | 3840 1187541 67018752 1525312500 19588521600 ... %e A376790 ... %o A376790 (Python) %o A376790 s = "" %o A376790 for d in range(3, 11): %o A376790 for n in range(1, d-1): %o A376790 k = d-n %o A376790 s = s + str((k**(2*n) - (n* (k**(n+1))) + (n-1) * (k**n))//4) + ", " %o A376790 print(s) %Y A376790 Cf. A100575 (column 2). %K A376790 nonn,tabl %O A376790 1,5 %A A376790 _Ryota Inagaki_, _Tanya Khovanova_, and _Austin Luo_, Oct 04 2024