A298592 Triangle read by rows: T(n,k) = number of parking functions of length n whose lead number is k.

1, 2, 1, 8, 5, 3, 50, 34, 25, 16, 432, 307, 243, 189, 125, 4802, 3506, 2881, 2401, 1921, 1296, 65536, 48729, 40953, 35328, 30208, 24583, 16807, 1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144, 20000000, 15217031, 13119879, 11708091, 10546875, 9453125, 8291909, 6880121, 4782969

Offset

1,2

Links

Table of n, a(n) for n=1..45.

D. Foata and J. Riordan,  Mappings of acyclic and parking functions, J. Aeq. Math., 10 (1974) 10-22.

Formula

T(n,k) = Sum_{j=k..n} binomial(n-1, j-1)*j^(j-2)*(n+1-j)^(n-1-j).

T(n,k) = A298593(n,k)/n.

T(n,k) = Sum_{j=k..n} A298594(n,j).

T(n,k) = (Sum_{j=k..n} A298597(n,j))/n.

Sum_{k=1..n} T(n,k) = A000272(n+1).

Example

Triangle begins:
        1;
        2,      1;
        8,      5,      3;
       50,     34,     25,     16;
      432,    307,    243,    189,    125;
     4802,   3506,   2881,   2401,   1921,   1296;
    65536,  48729,  40953,  35328,  30208,  24583,  16807;
  1062882, 800738, 683089, 601441, 531441, 461441, 379793, 262144;
  ...

Mathematica

Table[Sum[Binomial[n - 1, j - 1] j^(j - 2)*(n + 1 - j)^(n - 1 - j), {j, k, n}], {n, 9}, {k, n}] // Flatten (* Michael De Vlieger, Jan 22 2018 *)

Crossrefs

Cf. A000272, A298593, A298594, A298597.

Sequence in context: A136230 A193892 A193907 * A344163 A199468 A337684

Adjacent sequences: A298589 A298590 A298591 * A298593 A298594 A298595

Keyword

easy,nonn,tabl

Author

Rui Duarte, Jan 22 2018

Status

approved