A391488 Triangle read by rows: T(n,k) is the number of endofunctions on [n] where the difference between the smallest and largest values in the image of the function is k, 0 <= k < n.

1, 2, 2, 3, 12, 12, 4, 42, 100, 110, 5, 120, 540, 1140, 1320, 6, 310, 2408, 8106, 16324, 19502, 7, 756, 9660, 48552, 142632, 280140, 341796, 8, 1778, 36300, 263350, 1064456, 2891706, 5592388, 6927230, 9, 4080, 130620, 1339740, 7242600, 25734360, 66454020, 127176420, 159338640

Offset

1,2

Links

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (rows 1..50)

Formula

T(n,0) = n.

T(n,k) = (n-k)*((k+1)^n - 2*k^n + (k-1)^n) for k > 0.

Example

Triangle begins:
  1;
  2,    2;
  3,   12,    12;
  4,   42,   100,    110;
  5,  120,   540,   1140,    1320;
  6,  310,  2408,   8106,   16324,   19502;
  7,  756,  9660,  48552,  142632,  280140,  341796;
  8, 1778, 36300, 263350, 1064456, 2891706, 5592388, 6927230;
  ...

Mathematica

A391488[n_, k_] := If[k == 0, n, (n-k)*((k+1)^n - 2*k^n + (k-1)^n)];
Table[A391488[n, k], {n, 10}, {k, 0, n-1}] (* Paolo Xausa, Jan 23 2026 *)

Prog

(PARI) T(n, k) = if(k==0, n, (n-k)*((k+1)^n - 2*k^n + (k-1)^n))

Crossrefs

Row sums are A000312.

Main diagonal is A126778, n > 1.

Column k=0 is A000027.

Sequence in context: A199673 A377660 A375218 * A335942 A240133 A293445

Adjacent sequences: A391485 A391486 A391487 * A391489 A391490 A391491

Keyword

nonn,tabl

Author

Andrew Howroyd, Jan 22 2026

Status

approved