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%I #18 Apr 06 2024 15:08:33
%S 1,0,1,8,0,0,36,0,180,0,49,0,1604,0,21,8064,0,5144,0,0,112608,0,7680,
%T 0,604800,0,604428,0,5445,0,11799360,0,1669052,0,1485,68428800,0,
%U 91705536,0,2610608,0,0,1741669632,0,384036016,0,2342340,0,10897286400,0,18071744976,0,972895560,0,1126125
%N 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.
%C T(n,k) is zero unless k <= n/2. (proven to be correct)
%H Robert Cori and Gábor Hetyei, <a href="https://arxiv.org/abs/2403.19569">On reduced unicellular hypermonopoles</a>, arXiv:2403.19569 [math.CO], 2024.
%F 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.
%F T(2*m+1,1) = (2*m)! / (m+1) = A060593(m) for m >= 1.
%e The table begins:
%e 1;
%e 0, 1;
%e 8, 0;
%e 0, 36, 0;
%e 180, 0, 49;
%e 0, 1604, 0, 21;
%e 8064, 0, 5144, 0;
%e 0, 112608, 0, 7680, 0;
%e 604800, 0, 604428, 0, 5445;
%e 0, 11799360, 0, 1669052, 0, 1485;
%e 68428800, 0, 91705536, 0, 2610608, 0;
%e 0, 1741669632, 0, 384036016, 0, 2342340, 0;
%p proc(n, k)
%p local i;
%p 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);
%p end proc
%Y Cf. A060593, A164652.
%K nonn,tabf
%O 3,4
%A _Gabor Hetyei_, Apr 02 2024
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