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 A320531 #4 Oct 18 2018 03:05:21
%S A320531 0,1,0,0,1,0,0,2,1,0,0,3,4,1,0,0,4,12,6,1,0,0,5,32,27,8,1,0,0,6,80,
%T A320531 108,48,10,1,0,0,7,192,405,256,75,12,1,0,0,8,448,1458,1280,500,108,14,
%U A320531 1,0,0,9,1024,5103,6144,3125,864,147,16,1,0,0,10,2304
%N A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.
%C A320531 T(n,k) is the number of length n*k binary words of n consecutive blocks of length k, respectively, one of the blocks having exactly k letters 1, and the other having exactly one letter 0. First column follows from the next definition.
%C A320531 In Kauffman's language, T(n,k) is the total number of Jordan trails that are obtained by placing state markers at the crossings of the Pretzel universe P(k, k, ..., k) having n tangles, of k half-twists respectively. In other words, T(n,k) is the number of ways of splitting the crossings of the Pretzel knot shadow P(k, k, ..., k) such that the final diagram is a single Jordan curve. The aforementionned binary words encode these operations by assigning each tangle a length k binary words with the adequate choice for splitting the crossings.
%C A320531 Columns are linear recurrence sequences with signature (2*k, -k^2).
%D A320531 Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.
%H A320531 Louis H. Kauffman, <a href="https://doi.org/10.1016/0040-9383(87)90009-7">State models and the Jones polynomial</a>, Topology, Vol. 26 (1987), 395-407.
%H A320531 Franck Ramaharo, <a href="https://arxiv.org/abs/1805.10680">A generating polynomial for the pretzel knot</a>, arXiv:1805.10680 [math.CO], 2018.
%H A320531 Alexander Stoimenow, <a href="https://doi.org/10.3390/sym7020365">Everywhere Equivalent 2-Component Links</a>, Symmetry Vol. 7 (2015), 365-375.
%H A320531 Wikipedia, <a href="https://en.wikipedia.org/wiki/Pretzel_link">Pretzel link</a>
%F A320531 T(n,k) = (2*k)*T(n-1,k) - (k^2)*T(n-2,k).
%F A320531 G.f. for columns: x/(1 - k*x)^2.
%F A320531 E.g.f. for columns: x*exp(k*x).
%F A320531 T(n,1) = A001477(n).
%F A320531 T(n,2) = A001787(n).
%F A320531 T(n,3) = A027471(n+1).
%F A320531 T(n,4) = A002697(n).
%F A320531 T(n,5) = A053464(n).
%F A320531 T(n,6) = A053469(n), n > 0.
%F A320531 T(n,7) = A027473(n), n > 0.
%F A320531 T(n,8) = A053539(n).
%F A320531 T(n,9) = A053540(n), n > 0.
%F A320531 T(n,10) = A053541(n), n > 0.
%F A320531 T(n,11) = A081127(n).
%F A320531 T(n,12) = A081128(n).
%e A320531 Square array begins:
%e A320531 0, 0, 0, 0, 0, 0, 0, 0, ...
%e A320531 1, 1, 1, 1, 1, 1, 1, 1, ...
%e A320531 0, 2, 4, 6, 8, 10, 12, 14, ... A005843
%e A320531 0, 3, 12, 27, 48, 75, 108, 147, ... A033428
%e A320531 0, 4, 32, 108, 256, 500, 864, 1372, ... A033430
%e A320531 0, 5, 80, 405, 1280, 3125, 6480, 12005, ... A269792
%e A320531 0, 6, 192, 1458, 6144, 18750, 46656, 100842, ...
%e A320531 0, 7, 448, 5103, 28672, 109375, 326592, 823543, ...
%e A320531 ...
%e A320531 T(3,2) = 3*2^(3 - 1) = 12. The corresponding binary words are 110101, 110110, 111001, 111010, 011101, 011110, 101101, 101110, 010111, 011011, 100111, 101011.
%t A320531 T[n_, k_] = If [k > 0, n*k^(n - 1), If[k == 0 && n == 1, 1, 0]];
%t A320531 Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 12}]//Flatten
%o A320531 (Maxima)
%o A320531 T(n, k) := if k > 0 then n*k^(n - 1) else if k = 0 and n = 1 then 1 else 0$
%o A320531 tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, nn))$
%Y A320531 Antidiagonal sums: A101495.
%Y A320531 Column 1 is column 2 of A300453.
%Y A320531 Column 2 is column 1 of A300184.
%Y A320531 Cf. A104002, A320530.
%K A320531 nonn,easy,tabl
%O A320531 0,8
%A A320531 _Franck Maminirina Ramaharo_, Oct 14 2018