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A344112
Triangle read by rows: T(n,k) is the number of relations from an n-element set to a k-element set that are not functions.
6
1, 3, 12, 7, 56, 485, 15, 240, 4015, 65280, 31, 992, 32525, 1047552, 33551307, 63, 4032, 261415, 16773120, 1073726199, 68719430080, 127, 16256, 2094965, 268419072, 34359660243, 4398046231168, 562949952597769, 255, 65280, 16770655, 4294901760, 1099511237151
OFFSET
1,2
LINKS
Michael De Vlieger, Table of n, a(n) for n = 1..1275 (rows n = 1..50, flattened)
Mohammad K. Azarian, Remarks and Conjectures Regarding Combinatorics of Discrete Partial Functions, Int'l Math. Forum (2022) Vol. 17, No. 3, 129-141.
FORMULA
T(n,k) = 2^(n*k) - k^n, n,k >= 1.
EXAMPLE
T(2,2) = (number of relations) - (number of functions) = 2^4 - 4 = 12.
Triangle T(n,k) begins:
1;
3, 12;
7, 56, 485;
15, 240, 4015, 65280;
31, 992, 32525, 1047552, 33551307;
MATHEMATICA
Column[Table[2^(n*k) - k^n, {n, 10}, {k, n}], Center]
CROSSREFS
KEYWORD
easy,nonn,tabl
AUTHOR
Mohammad K. Azarian, May 10 2021
STATUS
approved