A260693 -id:A260693 - cp's OEIS frontend

A291121 Number of parking functions of length 2n whose maximum element is n.

1, 1, 14, 596, 51570, 7499467, 1643722873, 505862470010, 207976214329922, 110082566480231855, 72903763226014970781, 59052859112474311283157, 57431375668375876890521461, 66037761629358556242334308571, 88619376126543463987900902115279

Offset: 0

Alois P. Heinz, Aug 17 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..214

Cf. A260693.

a(n) = A260693(2n,n).

A373086 Triangle read by rows: T(n, k) is the number of parking functions of length n with preferences restricted to {1, ..., k} for 0 <= k <= n.

Original entry on oeis.org

1, 0, 1, 0, 1, 3, 0, 1, 7, 16, 0, 1, 15, 61, 125, 0, 1, 31, 206, 671, 1296, 0, 1, 63, 659, 3130, 9031, 16807, 0, 1, 127, 2052, 13686, 54062, 144495, 262144, 0, 1, 255, 6297, 57867, 301321, 1059261, 2685817, 4782969, 0, 1, 511, 19162, 240049, 1616764, 7196785, 23343742, 56953279, 100000000

Offset: 0

Jayden Thadani and Alan Kappler, May 23 2024

Equivalently, this is the number of preference lists of n cars on k spots so that only n - k cars are unable to park.

			Table begins:
  1;
  0, 1;
  0, 1,   3;
  0, 1,   7,    16;
  0, 1,  15,    61,    125;
  0, 1,  31,   206,    671,    1296;
  0, 1,  63,   659,   3130,    9031,   16807;
  0, 1, 127,  2052,  13686,   54062,  144495,   262144;
  0, 1, 255,  6297,  57867,  301321, 1059261,  2685817,  4782969;
  0, 1, 511, 19162, 240049, 1616764, 7196785, 23343742, 56953279, 100000000;
  ...

P. J. Cameron, D. Johannsen, T. Prellberg, and P. Schweitzer, Counting Defective Parking Functions, Electronic Journal of Combinatorics, Volume 15 (2008).

Cf. A000272 (diagonal), A000225 (column 2), A355645 (column 3, with offset 3).

Cf. A260693 (first differences of columns).

T(n, k) = k^n - Sum_{i=0..k-1} binomial(n, i)*(i+1)^(i-1)*(k-i-1)^(n-i).

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

cp's OEIS Frontend

A291121 Number of parking functions of length 2n whose maximum element is n.

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