A320530 T(n,k) = k^n + k^(n - 2)*n*(n - 1)*(k*(k - 1) + 1)/2 for 0 < k <= n and T(n,0) = A154272(n+1), square array read by antidiagonals upwards.
1, 1, 1, 0, 1, 1, 0, 2, 2, 1, 0, 4, 7, 3, 1, 0, 7, 26, 16, 4, 1, 0, 11, 88, 90, 29, 5, 1, 0, 16, 272, 459, 220, 46, 6, 1, 0, 22, 784, 2133, 1504, 440, 67, 7, 1, 0, 29, 2144, 9234, 9344, 3775, 774, 92, 8, 1, 0, 37, 5632, 37908, 54016, 29375, 7992, 1246, 121, 9
Offset: 0
Examples
Square array begins:
1, 1, 1, 1, 1, 1, 1, ...
0, 1, 2, 3, 4, 5, 6, ...
1, 2, 7, 16, 29, 46, 67, ...
0, 4, 26, 90, 220, 440, 774, ...
0, 7, 88, 459, 1504, 3775, 7992, ...
0, 11, 272, 2133, 9344, 29375, 74736, ...
0, 16, 784, 9234, 54016, 212500, 649296, ...
0, 22, 2144, 37908, 295936, 1456250, 5342112, ...
...
T(3,2) = 2^3 + 2^(3 - 2)*3*(3 - 1)*(2*(2 - 1) + 1)/2 = 26. The corresponding ternary words are abc, acb, cab, bac, bca, cba, bbc, bcb, cbb, ccc. Next, let a = {00}, b = {11} and c = {01, 10}. The resulting binary words are
abc: 001101, 001110;
acb: 000111, 001011;
cab: 010011, 100011;
bac: 110001, 110010;
bca: 110100, 111000;
cba: 011100, 101100;
bbc: 111101, 111110;
bcb: 110111, 111011;
cbb: 011111, 101111;
ccc: 010101, 101010, 010110, 011001, 100101, 101001, 100110, 011010.
References
- Louis H. Kauffman, Formal Knot Theory, Princeton University Press, 1983.
Links
- 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
Crossrefs
Programs
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Mathematica
T[n_, k_] = If[k > 0, k^n + k^(n - 2)*n*(n - 1)*(k*(k - 1) + 1)/2, If[k == 0 && (n == 0 || n == 1), 1, 0]]; Table[Table[T[n - k, k], {k, 0, n}], {n, 0, 10}]//Flatten -
Maxima
t(n, k) := k^n + k^(n - 2)*binomial(n, 2)*(2*binomial(k, 2) + 1)$ u(n) := if n = 0 or n = 1 then 1 else 0$ T(n, k) := if k = 0 then u(n) else t(n,k)$ tabl(nn) := for n:0 thru 10 do print(makelist(T(n, k), k, 0, nn))$
Formula
T(n,k) = k^n + k^(n - 2)*binomial(n, 2)*(2*binomial(k, 2) + 1), k > 0.
T(n,k) = (3*k)*T(n-1,k) - (3*k^2)*T(n-2,k) + (k^3)*T(n-3,k), n > 3.
T(n,1) = A152947(n+1).
T(n,2) = A300451(n).
T(2,n) = A130883(n).
G.f. for columns: (1 - 2*k*x + (1 - k + 2*k^2)*x^2 )/(1 - k*x)^3.
E.g.f. for columns: ((1 - k + k^2)*x^2 + 2)*exp(k*x)/2.
Comments