A320531 T(n,k) = n*k^(n - 1), k > 0, with T(n,0) = A063524(n), square array read by antidiagonals upwards.

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, 108, 48, 10, 1, 0, 0, 7, 192, 405, 256, 75, 12, 1, 0, 0, 8, 448, 1458, 1280, 500, 108, 14, 1, 0, 0, 9, 1024, 5103, 6144, 3125, 864, 147, 16, 1, 0, 0, 10, 2304

Offset

0,8

Comments

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.

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.

Columns are linear recurrence sequences with signature (2*k, -k^2).

References

Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.

Links

Table of n, a(n) for n=0..68.

Louis H. Kauffman, State models and the Jones polynomial, Topology, Vol. 26 (1987), 395-407.

Franck Ramaharo, A generating polynomial for the pretzel knot, arXiv:1805.10680 [math.CO], 2018.

Alexander Stoimenow, Everywhere Equivalent 2-Component Links, Symmetry Vol. 7 (2015), 365-375.

Wikipedia, Pretzel link

Formula

T(n,k) = (2*k)*T(n-1,k) - (k^2)*T(n-2,k).

G.f. for columns: x/(1 - k*x)^2.

E.g.f. for columns: x*exp(k*x).

T(n,1) = A001477(n).

T(n,2) = A001787(n).

T(n,3) = A027471(n+1).

T(n,4) = A002697(n).

T(n,5) = A053464(n).

T(n,6) = A053469(n), n > 0.

T(n,7) = A027473(n), n > 0.

T(n,8) = A053539(n).

T(n,9) = A053540(n), n > 0.

T(n,10) = A053541(n), n > 0.

T(n,11) = A081127(n).

T(n,12) = A081128(n).

Example

Square array begins:
    0, 0,   0,    0,     0,      0,      0,      0, ...
    1, 1,   1,    1,     1,      1,      1,      1, ...
    0, 2,   4,    6,     8,     10,     12,     14, ... A005843
    0, 3,  12,   27,    48,     75,    108,    147, ... A033428
    0, 4,  32,  108,   256,    500,    864,   1372, ... A033430
    0, 5,  80,  405,  1280,   3125,   6480,  12005, ... A269792
    0, 6, 192, 1458,  6144,  18750,  46656, 100842, ...
    0, 7, 448, 5103, 28672, 109375, 326592, 823543, ...
    ...
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.

Mathematica

T[n_, k_] = If [k > 0, n*k^(n - 1), If[k == 0 && n == 1, 1, 0]];
Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 12}]//Flatten

Prog

(Maxima)
T(n, k) := if k > 0 then n*k^(n - 1) else if k = 0 and n = 1 then 1 else 0$
tabl(nn) := for n:0 thru nn do print(makelist(T(n, k), k, 0, nn))$

Crossrefs

Antidiagonal sums: A101495.

Column 1 is column 2 of A300453.

Column 2 is column 1 of A300184.

Cf. A104002, A320530.

Sequence in context: A221984 A071920 A306548 * A345698 A369195 A065719

Adjacent sequences: A320528 A320529 A320530 * A320532 A320533 A320534

Keyword

nonn,easy,tabl

Author

Franck Maminirina Ramaharo, Oct 14 2018

Status

approved