A387937 Triangle read by rows: T(n, k) is the number of cycles of length n with k descents in a complete graph on n nodes.

1, 1, 1, 1, 19, 1, 1, 101, 101, 1, 1, 426, 1751, 426, 1, 1, 1667, 18327, 18327, 1667, 1, 1, 6371, 154645, 398567, 154645, 6371, 1, 1, 24229, 1170073, 6262237, 6262237, 1170073, 24229, 1, 1, 92278, 8337880, 82044082, 167729959, 82044082, 8337880, 92278, 1

Offset

2,5

Comments

If M is the n X n matrix filled with a variable t above the diagonal, one below the diagonal and zero on the diagonal, row(n) is the upper left element of M^n.

Every row is palindromic.

This should also give the primitive Hodge numbers of Dwork hypersurfaces.

Links

Table of n, a(n) for n=2..46.

Example

As a triangle:
  [1],
  [1, 1],
  [1, 19, 1],
  [1, 101, 101, 1],
  [1, 426, 1751, 426, 1],
  [1, 1667, 18327, 18327, 1667, 1]

Prog

(SageMath)
t = polygen(ZZ, 't')
def M(n):
    return ((matrix(n, n, lambda i, j: t if i < j else 1) - 1)**n)[0, 0]
[list(M(i))[1:] for i in range(2, 8)]
(PARI) row(n) = my(m=matrix(n, n, i, j, if (i>j, 1, if (i<j, t, 0)))); Vec((m^n)[1, 1], n-1); \\ Michel Marcus, Sep 15 2025

Crossrefs

Row sums are A185634. Cf. A387913.

Sequence in context: A040361 A174097 A174040 * A176078 A022182 A015145

Adjacent sequences: A387934 A387935 A387936 * A387938 A387939 A387940

Keyword

nonn,tabl

Author

F. Chapoton, Sep 13 2025

Status

approved