OFFSET
0,6
COMMENTS
S(n,x) is the sum of subword complexities (number of nonempty distinct subwords) of all words of length n and an alphabet with size x.
Note that although the coefficients can be negative, S(n,x) is always a nonnegative number for n,x >= 0.
The degree of S(n,x) is n.
The constant coefficient of S(n,x) is always 0.
Conjecture: the coefficient of x^n in S(n,x) is n*(n+1)/2.
LINKS
Shiyao Guo, Table of n, a(n) for n = 0..1890
Shiyao Guo, On the Expected Subword Complexity of Random Words.
EXAMPLE
The triangle begins as
0;
0, 1;
0, -1, 3;
0, 0, -3, 6;
0, -1, 1, -6, 10;
0, 2, -6, 4, -10, 15;
0, -2, 10, -18, 10, -15, 21;
0, 2, -12, 31, -41, 20, -21, 28;
...
Below lists some subword complexity polynomials:
S(0,x) = 0
S(1,x) = 1*x
S(2,x) = -1*x + 3*x^2
S(3,x) = -3*x^2 + 6*x^3
S(4,x) = -1*x + x^2 - 6*x^3 + 10*x^4
...
For n = 3 and x = 2 there are eight possible words: "aaa", "aab", "aba", "abb", "baa", "bab", "bba" and "bbb", and their subword complexities are 3, 5, 5, 5, 5, 5, 5 and 3 respectively, and their sum = S(3,2) = -3*(2^2)+6*(2^3) = 36.
MATHEMATICA
S[n_, x_] := Total[Length /@ DeleteDuplicates /@ Subsequences /@ Tuples[Table[i, {i, 0, x}], n] - 1]; A341761[n_] := CoefficientList[FindSequenceFunction[ParallelTable[S[n, i], {i, 0, n + 1}], x], {x}]; Join[{0, 0, 1}, Table[A341761[n], {n, 2, 7}] // Flatten] (* Robert P. P. McKone, Feb 20 2021 *)
PROG
(C++) // see link above
CROSSREFS
KEYWORD
sign,tabl
AUTHOR
Shiyao Guo, Feb 19 2021
STATUS
approved