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.

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

Author

Gabor Hetyei, Apr 02 2024

Keywords

• nonn
• tabf

Comments

T(n,k) is zero unless k <= n/2. (proven to be correct)

Examples

			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;
		

Links

• Robert Cori and Gábor Hetyei, On reduced unicellular hypermonopoles, arXiv:2403.19569 [math.CO], 2024.

Crossrefs

Cf. A060593, A164652.

Programs

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

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.