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

3,4

Comments

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

Links

Table of n, a(n) for n=3..57.

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

Cf. A060593, A164652.

Sequence in context: A340915 A028701 A126270 * A169696 A192059 A191419

Adjacent sequences: A371662 A371663 A371664 * A371666 A371667 A371668

Keyword

nonn,tabf

Author

Gabor Hetyei, Apr 02 2024

Status

approved