OFFSET
0,3
COMMENTS
If you take the powers of a finite function you generate a lollipop graph. This table organizes the lollipops by cycle size. The table organized by total lollipop size with the tail included is A225725.
Warning: For T(n,k) after the sixth row there are zero entries and k can be greater than n: T(7,k) = |{1=>262144, 2=>292383, 3=>145320, 4=>71610, 5=>24192, 6=>26250, 7=>720, 8=>0, 9=>0, 10=>504, 11=>0, 12=>420}|.
LINKS
Alois P. Heinz, Rows n = 0..30, flattened
Chad Brewbaker, Ruby program for A222029
FORMULA
EXAMPLE
T(1,1) = |{[0]}|, T(2,1) = |{[0,0],[0,1],[1,1]}|, T(2,2) = |{[0,1]}|.
Triangle starts:
1;
1;
3, 1;
16, 9, 2;
125, 93, 32, 6;
1296, 1155, 480, 150, 24, 20;
16807, 17025, 7880, 3240, 864, 840;
262144, 292383, 145320, 71610, 24192, 26250, 720, 0, 0, 504, 0, 420;
...
MAPLE
b:= proc(n, m) option remember; `if`(n=0, x^m, add((j-1)!*
b(n-j, ilcm(m, j))*binomial(n-1, j-1), j=1..n))
end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(add(
b(j, 1)*n^(n-j)*binomial(n-1, j-1), j=0..n)):
seq(T(n), n=0..10); # Alois P. Heinz, Aug 14 2017
MATHEMATICA
b[n_, m_]:=b[n, m]=If[n==0, x^m, Sum[(j - 1)!*b[n - j, LCM[m, j]] Binomial[n - 1, j - 1], {j, n}]]; T[n_]:=If[n==0, {1}, Drop[CoefficientList[Sum[b[j, 1]n^(n - j)*Binomial[n - 1, j - 1], {j, 0, n}], x], 1]]; Table[T[n], {n, 0, 10}]//Flatten (* Indranil Ghosh, Aug 17 2017 *)
PROG
#(Ruby 1.9+) see link.
(Python)
from sympy.core.cache import cacheit
from sympy import binomial, Symbol, lcm, factorial as f, Poly, flatten
x=Symbol('x')
@cacheit
def b(n, m): return x**m if n==0 else sum([f(j - 1)*b(n - j, lcm(m, j))*binomial(n - 1, j - 1) for j in range(1, n + 1)])
def T(n): return Poly(sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in range(n + 1)]), x).all_coeffs()[::-1][1:]
print([T(n) for n in range(11)]) # Indranil Ghosh, Aug 17 2017
CROSSREFS
KEYWORD
AUTHOR
Chad Brewbaker, May 14 2013
EXTENSIONS
T(0,1)=1 prepended by Alois P. Heinz, Aug 14 2017
STATUS
approved