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A371665
T(n,k) is the number of reduced unicellular hypermonopoles on n points with k hyperedges, where T(n,k), 1 <= k <= floor(n/2), is an array read by rows.
0
1, 0, 1, 8, 0, 0, 36, 0, 180, 0, 49, 0, 1604, 0, 21, 8064, 0, 5144, 0, 0, 112608, 0, 7680, 0, 604800, 0, 604428, 0, 5445, 0, 11799360, 0, 1669052, 0, 1485, 68428800, 0, 91705536, 0, 2610608, 0, 0, 1741669632, 0, 384036016, 0, 2342340, 0, 10897286400, 0, 18071744976, 0, 972895560, 0, 1126125
OFFSET
3,4
COMMENTS
T(n,k) is zero unless k <= n/2. (proven to be correct)
LINKS
Robert Cori and Gábor Hetyei, On reduced unicellular hypermonopoles, arXiv:2403.19569 [math.CO], 2024.
FORMULA
T(n,k) = Sum_{i=0..k-1} (-1)^i binomial(n,i)*a(n-1-i,k-i) where the a(n,k) are the Hultman numbers from A164652.
T(2*m+1,1) = (2*m)! / (m+1) = A060593(m) for m >= 1.
EXAMPLE
The table begins:
1;
0, 1;
8, 0;
0, 36, 0;
180, 0, 49;
0, 1604, 0, 21;
8064, 0, 5144, 0;
0, 112608, 0, 7680, 0;
604800, 0, 604428, 0, 5445;
0, 11799360, 0, 1669052, 0, 1485;
68428800, 0, 91705536, 0, 2610608, 0;
0, 1741669632, 0, 384036016, 0, 2342340, 0;
MAPLE
proc(n, k)
local i;
coeff(expand(add(combinat:-binomial(n, i)*(-x)^i*(pochhammer(x, n - i + 1) - pochhammer(x - n + i, n - i + 1))/((n - i)*(n - i + 1)), i = 0 .. n - 1)), x, k);
end proc
CROSSREFS
Sequence in context: A340915 A028701 A126270 * A169696 A192059 A191419
KEYWORD
nonn,tabf
AUTHOR
Gabor Hetyei, Apr 02 2024
STATUS
approved