A264902 -id:A264902 - cp's OEIS frontend

A291132 Number of defective parking functions of length n and defect six.

1, 303, 34660, 2743112, 181875244, 11023248678, 639875755364, 36555471741284, 2090131479753756, 120898503338385149, 7124218746544184628, 429662666436736162636, 26601747798152634836236, 1694092238645618305809580, 111106187207006959809867012

Offset: 7

Alois P. Heinz, Aug 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 7..386
Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Column k=6 of A264902.

S:= (n, k)-> add(binomial(n, i)*k*(k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k):
a:= n-> S(n, 6)-S(n, 7):
seq(a(n), n=7..23);

S[n_, k_] := Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}];
a[n_] := S[n, 6] - S[n, 7];
Table[a[n], {n, 7, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

a(n) ~ (43*exp(1)/720 - 88*exp(2)/15 + 405*exp(3)/8 - 368*exp(4)/3 + 235*exp(5)/2 - 48*exp(6) + 7*exp(7)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017

A291133 Number of defective parking functions of length n and defect seven.

Original entry on oeis.org

1, 574, 96620, 10358998, 886044810, 66943181150, 4719570364004, 320771944968342, 21454694483447459, 1431385710008667470, 96133394595460111056, 6540549310477955461846, 452777288307033641080180, 31990399760398854681388158, 2311790354938282481939931160

Offset: 8

Alois P. Heinz, Aug 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 8..386
Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Column k=7 of A264902.

S:= (n, k)-> add(binomial(n, i)*k*(k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k):
a:= n-> S(n, 7)-S(n, 8):
seq(a(n), n=8..23);

S[n_, k_] := Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}];
a[n_] := S[n, 7] - S[n, 8];
Table[a[n], {n, 8, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

a(n) ~ (-19*exp(1)/1680 + 116*exp(2)/45 - 1593*exp(3)/40 + 160*exp(4) - 1525*exp(5)/6 + 186*exp(6) - 63*exp(7) + 8*exp(8)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017

A291134 Number of defective parking functions of length n and defect eight.

Original entry on oeis.org

1, 1103, 269512, 38643849, 4218834608, 393933602129, 33499946915016, 2693983725947891, 209859823775671984, 16093162912317174422, 1228462028909579534968, 94081283153407041089350, 7269699339591280955315232, 569088494101518607733459806

Offset: 9

Alois P. Heinz, Aug 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 9..386
Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Column k=8 of A264902.

S:= (n, k)-> add(binomial(n, i)*k*(k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k):
a:= n-> S(n, 8)-S(n, 9):
seq(a(n), n=9..23);

S[n_, k_] := Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}];
a[n_] := S[n, 8] - S[n, 9];
Table[a[n], {n, 9, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

a(n) ~ (73*exp(1)/40320 - 296*exp(2)/315 + 405*exp(3)/16 - 2432*exp(4)/15 + 9625*exp(5)/24 - 468*exp(6) + 553*exp(7)/2 - 80*exp(8) + 9*exp(9)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017

A291135 Number of defective parking functions of length n and defect nine.

Original entry on oeis.org

1, 2146, 754943, 143336610, 19795924787, 2267392009178, 231141766226605, 21881366451890002, 1976997422623843358, 173666031731576614842, 15025473411620865716938, 1292364106829281911023554, 111260031164008673095102874, 9635674549219284395173044506

Offset: 10

Alois P. Heinz, Aug 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 10..386
Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Column k=9 of A264902.

S:= (n, k)-> add(binomial(n, i)*k*(k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k):
a:= n-> S(n, 9)-S(n, 10):
seq(a(n), n=10..23);

S[n_, k_] := Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}];
a[n_] := S[n, 9] - S[n, 10];
Table[a[n], {n, 10, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

a(n) ~ (-13*exp(1)/51840 + 92*exp(2)/315 - 7533*exp(3)/560 + 6016*exp(4)/45 - 11875*exp(5)/24 + 864*exp(6) - 4753*exp(7)/6 + 392*exp(8) - 99*exp(9) + 10*exp(10)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017

A291136 Number of defective parking functions of length n and defect ten.

Original entry on oeis.org

1, 4215, 2127828, 530926606, 92071525556, 12851428617547, 1561750852160556, 173226805226723844, 18081637592017744356, 1813499364725872444178, 177350996523515552397628, 17092810524840161845093436, 1636375630004710170560408532, 156537967540558397590739941650

Offset: 11

Alois P. Heinz, Aug 18 2017

nonn

Alois P. Heinz, Table of n, a(n) for n = 11..386
Peter J. Cameron, Daniel Johannsen, Thomas Prellberg, Pascal Schweitzer, Counting Defective Parking Functions, arXiv:0803.0302 [math.CO], 2008.

Column k=10 of A264902.

S:= (n, k)-> add(binomial(n, i)*k*(k+i)^(i-1)*(n-k-i)^(n-i), i=0..n-k):
a:= n-> S(n, 10)-S(n, 11):
seq(a(n), n=11..23);

S[n_, k_] := Sum[Binomial[n, i]*k*(k+i)^(i-1)*(n-k-i)^(n-i), {i, 0, n-k}];
a[n_] := S[n, 10] - S[n, 11];
Table[a[n], {n, 11, 23}] (* Jean-François Alcover, Feb 24 2019, from Maple *)

a(n) ~ (37*exp(1)/1209600 - 32*exp(2)/405 + 27459*exp(3)/4480 - 9728*exp(4)/105 + 71875*exp(5)/144 - 6264*exp(6)/5 + 13377*exp(7)/8 - 3776*exp(8)/3 + 1071*exp(9)/2 - 120*exp(10) + 11*exp(11)) * n^(n-1). - Vaclav Kotesovec, Aug 19 2017

A365623 T(n,k) is the number of parking functions of length n with cars parking at most k spots away from their preferred spot; square array T(n,k), n>=0, k>=0, read by downward antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 3, 6, 1, 1, 3, 13, 24, 1, 1, 3, 16, 75, 120, 1, 1, 3, 16, 109, 541, 720, 1, 1, 3, 16, 125, 918, 4683, 5040, 1, 1, 3, 16, 125, 1171, 9277, 47293, 40320, 1, 1, 3, 16, 125, 1296, 12965, 109438, 545835, 362880, 1, 1, 3, 16, 125, 1296, 15511, 166836, 1475691, 7087261, 3628800

Offset: 0

J. Carlos Martínez Mori, Sep 13 2023

			Square array T(n,k) begins:
    1,    1,    1,     1,     1,     1,     1, ...
    1,    1,    1,     1,     1,     1,     1, ...
    2,    3,    3,     3,     3,     3,     3, ...
    6,   13,   16,    16,    16,    16,    16, ...
   24,   75,  109,   125,   125,   125,   125, ...
  120,  541,  918,  1171,  1296,  1296,  1296, ...
  720, 4683, 9277, 12965, 15511, 16807, 16807, ...
  ...

Columns k=0..1, 3..4 give: A000142, A000670, A365626, A365627.
Main diagonal gives A000272(n+1).
Cf. A264902.

T:= proc(n, k) option remember; `if`(n=0, 1, add(min(i+1, k+1)*
       binomial(n-1, i)*T(i, k)*T(n-1-i, k), i=0..n-1))
    end:
seq(seq(T(n, d-n), n=0..d), d=0..10);  # Alois P. Heinz, Sep 13 2023

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

cp's OEIS Frontend

A291132 Number of defective parking functions of length n and defect six.

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A291133 Number of defective parking functions of length n and defect seven.

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A291134 Number of defective parking functions of length n and defect eight.

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A291135 Number of defective parking functions of length n and defect nine.

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A291136 Number of defective parking functions of length n and defect ten.

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A365623 T(n,k) is the number of parking functions of length n with cars parking at most k spots away from their preferred spot; square array T(n,k), n>=0, k>=0, read by downward antidiagonals.

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