OFFSET
0,4
LINKS
Andrew Howroyd, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (11,-55,165,-330,462,-462,330,-165,55,-11,1).
FORMULA
a(n) = 27*C(n,3) + 660*C(n,4) + 5201*C(n,5) + 21822*C(n,6) + 54600*C(n,7) + 84000*C(n,8) + 75600*C(n,9) + 30240*C(n,10).
a(n) = (1/120)*(n^10 - 20*n^9 + 220*n^8 - 1500*n^7 + 6710*n^6 - 19954*n^5 + 38765*n^4 - 46950*n^3 + 31944*n^2 - 9216*n).
a(n) = C(n,3)*(n^7 - 17*n^6 + 167*n^5 - 965*n^4 + 3481*n^3 - 7581*n^2 + 9060*n - 4608)/20. - Chai Wah Wu, Jan 06 2022
G.f.: x^3*(27 + 471*x + 1808*x^2 + 4772*x^3 + 7067*x^4 + 8455*x^5 + 6346*x^6 + 1294*x^7)/(1 - x)^11. - Andrew Howroyd, Nov 14 2025
EXAMPLE
No relation containing exactly five ordered pairs on a 2-element set exists. Thus a(2)=0.
Also, there are 27 transitive relations with exactly five ordered pairs on a 3-set. One such relation is {(1,1),(1,2),(1,3),(2,2),(3,2)} on the 3-set {1,2,3}.
MATHEMATICA
A348634[n_] := Total[{27, 660, 5201, 21822, 54600, 84000, 75600, 30240}*Binomial[n, Range[3, 10]]];
Array[A348634, 30, 0] (* or *)
LinearRecurrence[{11, -55, 165, -330, 462, -462, 330, -165, 55, -11, 1}, {0, 0, 0, 27, 768, 8771, 63468, 340620, 1470784, 5371002, 17153352}, 30] (* Paolo Xausa, Mar 24 2026 *)
PROG
(Python)
def A348634(n): return n*(n - 2)*(n - 1)*(n*(n*(n*(n*(n*(n*(n - 17) + 167) - 965) + 3481) - 7581) + 9060) - 4608)//120 # Chai Wah Wu, Jan 06 2022
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Firdous Ahmad Mala, Dec 13 2021
EXTENSIONS
a(9) corrected by Georg Fischer, Mar 19 2023
STATUS
approved