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 A222029 Triangle of number of functions in a size n set for which the sequence of composition powers ends in a length k cycle. 15
 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, 4782969, 5752131, 3009888, 1692180, 653184, 773920, 46080, 5040, 0, 32256, 0, 26880, 0, 0, 2688 (list; graph; refs; listen; history; text; internal format)
 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 Sum_{k=1..A000793(n)} k * T(n,k) = A290932. - Alois P. Heinz, Aug 14 2017 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 xrange(1, n + 1)]) def T(n): return Poly(sum([b(j, 1)*n**(n - j)*binomial(n - 1, j - 1) for j in xrange(n + 1)])).all_coeffs()[::-1][1:] print flatten(map(T, xrange(11))) # Indranil Ghosh, Aug 17 2017 CROSSREFS Columns k=1-10 give A000272, A163951, A163952, A291110, A291111, A291112, A291113, A291114, A291115, A291116. Rows sums give A000312. Row lengths are A000793. Number of nonzero elements of rows give A009490. Last elements of rows give A162682. Main diagonal gives A290961. Cf. A057731 (the same for permutations), A290932. Sequence in context: A102012 A128249 A071211 * A038675 A264902 A156653 Adjacent sequences:  A222026 A222027 A222028 * A222030 A222031 A222032 KEYWORD nonn,look,tabf AUTHOR Chad Brewbaker, May 14 2013 EXTENSIONS T(0,1)=1 prepended by Alois P. Heinz, Aug 14 2017 STATUS approved

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Last modified October 21 08:47 EDT 2019. Contains 328292 sequences. (Running on oeis4.)