login
A322013
Square array A(n,k), n >= 1, k >= 1, read by antidiagonals, where A(n,k) is the number of permutations of n copies of 1..k introduced in order 1..k with no element equal to another within a distance of 1.
11
1, 1, 0, 1, 1, 0, 1, 5, 1, 0, 1, 36, 29, 1, 0, 1, 329, 1721, 182, 1, 0, 1, 3655, 163386, 94376, 1198, 1, 0, 1, 47844, 22831355, 98371884, 5609649, 8142, 1, 0, 1, 721315, 4420321081, 182502973885, 66218360625, 351574834, 56620, 1, 0
OFFSET
1,8
LINKS
Evgeniy Krasko, Igor Labutin, and Alexander Omelchenko, Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete k-partite Graphs, arXiv:1709.03218 [math.CO], 2017.
Mathematics.StackExchange, Find the number of k 1's, k 2's, ... , k n's - total kn cards, Apr 08 2012.
FORMULA
T(n,k) = A322093(n,k) / k!. - Andrew Howroyd, Feb 03 2024
EXAMPLE
Square array begins:
1, 1, 1, 1, 1, 1, ...
0, 1, 5, 36, 329, 3655, ...
0, 1, 29, 1721, 163386, 22831355, ...
0, 1, 182, 94376, 98371884, 182502973885, ...
0, 1, 1198, 5609649, 66218360625, 1681287695542855, ...
0, 1, 8142, 351574834, 47940557125969, 16985819072511102549, ...
PROG
(PARI)
q(n, x) = sum(i=1, n, (-1)^(n-i) * binomial(n-1, n-i) * x^i/i!)
T(n, k) = subst(serlaplace(q(n, x)^k), x, 1)/k! \\ Andrew Howroyd, Feb 03 2024
CROSSREFS
Main diagonal gives A321666.
Sequence in context: A157012 A102365 A269945 * A102259 A021200 A265301
KEYWORD
nonn,tabl
AUTHOR
Seiichi Manyama, Nov 24 2018
STATUS
approved