A391488 - OEIS

%I #9 Jan 23 2026 08:44:39

%S 1,2,2,3,12,12,4,42,100,110,5,120,540,1140,1320,6,310,2408,8106,16324,

%T 19502,7,756,9660,48552,142632,280140,341796,8,1778,36300,263350,

%U 1064456,2891706,5592388,6927230,9,4080,130620,1339740,7242600,25734360,66454020,127176420,159338640

%N 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.

%H Andrew Howroyd, <a href="/A391488/b391488.txt">Table of n, a(n) for n = 1..1275</a> (rows 1..50)

%F T(n,0) = n.

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

%e Triangle begins:

%e   1;

%e   2,    2;

%e   3,   12,    12;

%e   4,   42,   100,    110;

%e   5,  120,   540,   1140,    1320;

%e   6,  310,  2408,   8106,   16324,   19502;

%e   7,  756,  9660,  48552,  142632,  280140,  341796;

%e   8, 1778, 36300, 263350, 1064456, 2891706, 5592388, 6927230;

%e   ...

%t A391488[n_, k_] := If[k == 0, n, (n-k)*((k+1)^n - 2*k^n + (k-1)^n)];

%t Table[A391488[n, k], {n, 10}, {k, 0, n-1}] (* _Paolo Xausa_, Jan 23 2026 *)

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

%Y Row sums are A000312.

%Y Main diagonal is A126778, n > 1.

%Y Column k=0 is A000027.

%K nonn,tabl

%O 1,2

%A _Andrew Howroyd_, Jan 22 2026